Question 1

A fraction is greater than twice its reciprocal by 7/15. What is the value of this fraction?

Accepted Answer

Introduction / Context: This problem again involves a relationship between a fraction and its reciprocal, but this time the fraction is greater than twice its reciprocal by a fixed amount. Such equations are a good test of algebraic translation skills and manipulation of quadratic equations. Mastering this pattern helps in many similar competitive exam problems. Given Data / Assumptions: Let the fraction be x (x ≠ 0). Its reciprocal is 1/x. We are told that x is greater than 2 * (1/x) by 7/15. So the equation is x - 2/x = 7/15. Concept / Approach: The given verbal condition is transformed into an algebraic equation involving x and 1/x. To solve it, we remove the denominator by multiplying through by x, giving a quadratic equation. Quadratic equations are then solved using the quadratic formula or factorization. Finally, we inspect the obtained roots and select the one that appears as an option and fits the context. Step-by-Step Solution: Step 1: Start from the equation: x - 2/x = 7/15. Step 2: Multiply both sides by x to eliminate the denominator: x * x - 2 = (7/15)x. Step 3: This gives x^2 - 2 = (7/15)x. Step 4: Multiply through by 15 to clear the fraction: 15x^2 - 30 = 7x. Step 5: Rearrange to standard form: 15x^2 - 7x - 30 = 0. Step 6: Apply the quadratic formula: x = [7 ± sqrt(7^2 - 4 * 15 * (-30))] / (2 * 15). Step 7: Compute the discriminant: 7^2 = 49; 4 * 15 * 30 = 1800; so discriminant = 49 + 1800 = 1849. Step 8: The square root of 1849 is 43, so x = (7 ± 43) / 30. Step 9: The roots are x = (7 + 43) / 30 = 50/30 = 5/3 and x = (7 - 43) / 30 = -36/30 = -6/5. Verification / Alternative Check: Test x = 5/3 in the original condition: x - 2/x = 5/3 - 2 * (3/5). Compute 2 * (3/5) = 6/5. Now 5/3 - 6/5 = (25/15) - (18/15) = 7/15, which matches the given difference. The other root, -6/5, also solves the equation but is not among the answer options and is negative, whereas the provided options show positive fractions. Therefore, 5/3 is the valid choice. Why Other Options Are Wrong: If x = 3/5, then x - 2/x will be negative, not 7/15. If x = 3/4 or x = 4/3, substituting into x - 2/x will not yield 7/15. Their differences from twice their reciprocals produce other values, so they do not satisfy the condition. Common Pitfalls: Students often misread the phrase “twice its reciprocal” and write x - 1/(2x) instead of x - 2/x. Another common mistake is sign error when moving terms across the equals sign. Careful algebra and clear writing of each step helps avoid these issues. Final Answer: The required fraction is 5/3.

Question 2

Which of the following fractions, when added to 5/8, gives a total sum equal to 1?

Accepted Answer

Introduction / Context: This question checks your understanding of fraction addition and complementary fractions that add up to 1. Many aptitude problems involve finding how much more is needed to complete a whole, a task, or a full unit. Here, you must find which fraction, when added to 5/8, results in exactly 1. Given Data / Assumptions: Known fraction: 5/8. Target sum: 1. We must find a fraction from the options so that 5/8 plus that fraction equals 1. Concept / Approach: If a/b + c/d = 1, then c/d is the complement of a/b to 1. The easiest way is to compute 1 - 5/8 and then match this result to one of the options. To subtract fractions, we use a common denominator. We may also express each option with denominator 8 to see if it matches the required complement. Step-by-Step Solution: Step 1: Compute the complement of 5/8 to 1: 1 - 5/8. Step 2: Write 1 as 8/8 to have the same denominator: 1 = 8/8. Step 3: Now subtract: 8/8 - 5/8 = (8 - 5) / 8 = 3/8. Step 4: So the required fraction is 3/8. Step 5: Check options to see which one is equal to 3/8. Step 6: Option A is 6/16. Simplify 6/16 by dividing numerator and denominator by 2: 6/16 = 3/8. Step 7: Therefore, 6/16 is equivalent to 3/8 and is the correct choice. Verification / Alternative Check: Add 5/8 and 6/16 directly. First, simplify 6/16 to 3/8. Then compute 5/8 + 3/8 = (5 + 3)/8 = 8/8 = 1. This confirms that adding 6/16 to 5/8 indeed gives a total of 1. Why Other Options Are Wrong: 5/2 is equal to 2.5, so 5/8 + 5/2 is much greater than 1. 6/24 simplifies to 1/4. Then 5/8 + 1/4 = 5/8 + 2/8 = 7/8, not 1. 6/3 is equal to 2, so 5/8 + 2 is greater than 2. Therefore none of these options give a sum of 1 when added to 5/8. Common Pitfalls: A common mistake is to mis-simplify fractions like 6/16 or 6/24. Always divide numerator and denominator by their greatest common divisor. Another error is to ignore simplification and compare fractions with different denominators directly, which is not reliable. Using a common denominator makes comparison and addition straightforward. Final Answer: The fraction that completes the sum to 1 is 6/16 (which is equal to 3/8).

Question 3

The sum of a fraction and three times its reciprocal is 31/6. What is the value of this fraction?

Accepted Answer

Introduction / Context: This problem is another example of an equation involving a fraction and its reciprocal. The fraction plus three times its reciprocal gives a specific value. Such expressions naturally lead to quadratic equations when we remove the denominator. Solving these equations efficiently is a key algebra skill for competitive exams. Given Data / Assumptions: Let the fraction be x (x ≠ 0). Its reciprocal is 1/x. We are given: x + 3 * (1/x) = 31/6. We must find x from the given answer choices. Concept / Approach: We express the verbal condition as an algebraic equation. Then we multiply through by x to clear the denominator, giving a quadratic equation in x. Using the quadratic formula or factorization, we obtain the roots. After that, we check which root appears among the options and suits the meaning of “fraction” in the context of the question. Step-by-Step Solution: Step 1: Start with the equation: x + 3/x = 31/6. Step 2: Multiply both sides by 6x to remove the denominator: 6x * x + 6x * (3/x) = 6x * 31/6. Step 3: Simplify: 6x^2 + 18 = 31x. Step 4: Rearrange into standard quadratic form: 6x^2 - 31x + 18 = 0. Step 5: Use the quadratic formula: x = [31 ± sqrt(31^2 - 4 * 6 * 18)] / (2 * 6). Step 6: Compute the discriminant: 31^2 = 961; 4 * 6 * 18 = 432; so discriminant = 961 - 432 = 529. Step 7: The square root of 529 is 23, so x = (31 ± 23) / 12. Step 8: The roots are x = (31 + 23) / 12 = 54/12 = 9/2 and x = (31 - 23) / 12 = 8/12 = 2/3. Verification / Alternative Check: Test x = 9/2 in the original expression. The reciprocal is 2/9. So compute x + 3/x = 9/2 + 3 * (2/9). First, 3 * (2/9) = 6/9 = 2/3. Now, 9/2 + 2/3 = (27/6) + (4/6) = 31/6, which matches the given value. The other root, 2/3, also satisfies the equation but is not listed among the options. Among the choices, 9/2 is the only correct match. Why Other Options Are Wrong: 2/9, 9/4 and 4/9 do not satisfy the equation when substituted. For example, with x = 9/4, the sum x + 3/x leads to a different value, not 31/6. Therefore, those options must be rejected. Common Pitfalls: One common error is failing to multiply through by the correct factor (here, 6x) to clear all denominators, which leads to an incorrect quadratic. Another pitfall is forgetting that a quadratic can have two valid roots; you still need to check the answer choices rather than assuming only one solution exists. Final Answer: The fraction that satisfies the condition is 9/2.

Question 4

Arrange the fractions 3/4, 5/12, 13/16, 16/29 and 3/8 in ascending order of their magnitude (from smallest to largest).

Accepted Answer

Introduction / Context: Ordering fractions from smallest to largest is a fundamental skill that helps in understanding ratios, proportions and comparisons in many quantitative problems. Here, five different fractions are given, and you must identify the option that lists them in ascending order of their magnitude. Given Data / Assumptions: Fractions: 3/4, 5/12, 13/16, 16/29 and 3/8. We must order them from smallest to largest. All fractions are positive and less than 1, except 13/16 and 3/4 which are close to 1. Concept / Approach: We can compare fractions using: Decimal conversion (approximate values). Cross-multiplication between pairs. Converting all fractions to a common denominator (though this may be more tedious here). In exam situations, converting to approximate decimals is often the fastest method when denominators are not too large. Step-by-Step Solution: Step 1: Convert each fraction into a decimal approximation. 3/8 = 0.375. 5/12 ≈ 5 ÷ 12 ≈ 0.4167. 16/29 ≈ 16 ÷ 29 ≈ 0.5517 (since 0.55 * 29 ≈ 15.95). 3/4 = 0.75 exactly. 13/16 = 0.8125. Step 2: List these decimal values: 0.375, 0.4167, 0.5517, 0.75, 0.8125. Step 3: Order them from smallest to largest: 0.375 < 0.4167 < 0.5517 < 0.75 < 0.8125. Step 4: Convert the ordered decimals back to fractions: 3/8 < 5/12 < 16/29 < 3/4 < 13/16. Verification / Alternative Check: We can verify individual pairwise comparisons using cross-multiplication. For example, to compare 3/8 and 5/12, check 3 * 12 = 36 and 5 * 8 = 40. Since 36 < 40, 3/8 < 5/12. Similar checks confirm the rest of the ordering, reinforcing our answer. Why Other Options Are Wrong: Option A: Places 3/4 before 3/8, which is wrong since 3/4 is much larger than 3/8. Option C: Incorrectly places 13/16 before 3/4, even though 13/16 is greater than 3/4 in actual value. Option D: Places 13/16 before 16/29 and 3/4, which contradicts the decimal comparisons. Only option B matches the correct ascending order. Common Pitfalls: A frequent error is to compare numerators or denominators separately without considering the fraction as a whole. Another mistake is miscalculating decimal approximations. Always check your arithmetic carefully, especially when dealing with non-terminating decimals, and be sure to maintain the correct order based on accurate comparisons. Final Answer: The correct ascending order is 3/8 < 5/12 < 16/29 < 3/4 < 13/16.

Question 5

When the recurring decimal 0.090909... is converted into a fraction in lowest terms, what is the resulting fraction?

Accepted Answer

Introduction / Context: Many competitive exams include questions on converting recurring (repeating) decimals into fractions. A recurring decimal has one or more digits that repeat in a fixed pattern. In this question, 0.090909... has the repeating block 09. We must convert this recurring decimal into a rational number in lowest terms. Given Data / Assumptions: Recurring decimal: 0.090909..., where the repeating block is 09. The repetition continues indefinitely. We must express this number as a fraction in simplest form. Concept / Approach: For a recurring decimal with a repeating block, we use the standard algebraic technique: Let x be the recurring decimal. Multiply x by a power of 10 that shifts one full repeat block to the left of the decimal point. Subtract the original x from this new expression to eliminate the repeating part. Solve for x as a fraction and simplify if needed. Step-by-Step Solution: Step 1: Let x = 0.090909... . Step 2: The repeating block is two digits (09), so multiply both sides by 100: 100x = 9.090909... . Step 3: Note that 100x and x share the same repeating part .090909..., which will cancel when we subtract. Step 4: Subtract the original equation from the new one: 100x - x = 9.090909... - 0.090909.... Step 5: This gives 99x = 9. Step 6: Solve for x: x = 9 / 99. Step 7: Simplify 9/99 by dividing numerator and denominator by 9: 9/99 = 1/11. Verification / Alternative Check: We know that 1/11 equals 0.090909... by standard fraction-decimal conversions. Dividing 1 by 11 produces 0.09 with 09 repeating forever. This confirms that our algebraic result is correct. Why Other Options Are Wrong: 1/33 gives a smaller repeating decimal approximately equal to 0.030303..., not 0.090909.... 2/33 corresponds to about 0.060606..., which still has a different value. 6/11 is approximately 0.545454..., whose repeating block is 54, not 09. Therefore, these options do not match the given decimal. Common Pitfalls: A common error is to incorrectly choose the power of 10, especially when the repeating block has two or more digits. Another mistake is simplifying the resulting fraction incorrectly. Always factor both numerator and denominator to remove any common divisor completely so that the answer is in lowest terms. Final Answer: The recurring decimal 0.090909... is equal to the fraction 1/11 in simplest form.

Question 6

When the recurring decimal 0.151515... is converted into a fraction in lowest terms, what is the resulting fraction?

Accepted Answer

Introduction / Context: This question requires converting another recurring decimal into a fraction. The decimal 0.151515... has the digit pair 15 repeating indefinitely. The goal is to express this repeating decimal as a rational number in its simplest form. Understanding this process is useful for many algebra and number system problems. Given Data / Assumptions: Recurring decimal: 0.151515... . The repeating block is 15. The sequence continues indefinitely. We must find an equivalent fraction in lowest terms. Concept / Approach: We use the standard algebraic method for repeating decimals: Let x equal the repeating decimal. Multiply x by a power of 10 such that the decimal point shifts beyond one full repeating block. Subtract the original x from this new multiple to eliminate the repeating part. Solve the resulting equation for x as a fraction and simplify. Step-by-Step Solution: Step 1: Let x = 0.151515... . Step 2: The repeating block (15) has two digits, so multiply both sides by 100: 100x = 15.151515... . Step 3: Note that 100x and x have the same repeating decimal part .151515..., which will cancel after subtraction. Step 4: Subtract the original equation from this new one: 100x - x = 15.151515... - 0.151515... . Step 5: This gives 99x = 15. Step 6: Solve for x: x = 15 / 99. Step 7: Simplify 15/99 by dividing numerator and denominator by their greatest common divisor 3: 15/99 = 5/33. Verification / Alternative Check: Check by converting 5/33 back to a decimal. Dividing 5 by 33 produces approximately 0.151515..., with 15 repeating indefinitely. This confirms that 5/33 is the correct fractional representation of the given decimal. Why Other Options Are Wrong: 17/33 corresponds to a larger fraction and gives approximately 0.515151... , not 0.151515.... 9/11 is about 0.818181..., with 81 repeating instead of 15. 7/33 produces approximately 0.212121..., again not matching the repeating block 15. Therefore, these fractions are not equivalent to the given decimal. Common Pitfalls: One common mistake is to treat 0.151515... as 15/100 plus a small remainder, rather than using the proper algebraic method. Another error is simplifying 15/99 incorrectly, for example stopping at 15/99 or dividing only by 9. Always divide by the greatest common divisor to reach lowest terms. Final Answer: The recurring decimal 0.151515... is equal to the fraction 5/33 in simplest form.

Question 7

Reduce the fraction 2714/5074 to its lowest terms (that is, express it in simplest fractional form).

Accepted Answer

Introduction / Context: Reducing fractions to lowest terms is a standard operation in arithmetic and is used repeatedly in algebra, ratio and proportion, and data interpretation. Here, we are given a relatively large fraction, 2714/5074, and we must simplify it by cancelling out common factors from the numerator and the denominator. Given Data / Assumptions: Original fraction: 2714/5074. We need to find an equivalent fraction that has the smallest possible positive numerator and denominator. The answer must match one of the options provided. Concept / Approach: To reduce a fraction to lowest terms, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD can be found by prime factorization, using the Euclidean algorithm, or testing divisibility by likely factors. Once we divide numerator and denominator by the GCD, the resulting fraction is in simplest form and will be unique. Step-by-Step Solution: Step 1: Consider the fraction 2714/5074. Step 2: Look for common factors between 2714 and 5074. Both numbers are even, so they are divisible by 2. Step 3: 2714 ÷ 2 = 1357 and 5074 ÷ 2 = 2537. So we can rewrite the fraction as 1357/2537. Step 4: Now look for a common factor between 1357 and 2537. Through further factor checking, we find that 2714 and 5074 actually share a greater common divisor, 118. Step 5: Divide numerator and denominator by 118 directly: 2714 ÷ 118 = 23 and 5074 ÷ 118 = 43. Step 6: The fraction simplifies exactly to 23/43. Step 7: Check if 23 and 43 have any common factor other than 1. Both are prime and distinct, so 23/43 is in lowest terms. Verification / Alternative Check: We can verify by multiplying back: 23 * 118 = 2714 and 43 * 118 = 5074. Since both are exact, 118 is indeed a common factor and 23/43 is the reduced fraction. No smaller common factor exists for 23 and 43, so we have the correct simplest form. Why Other Options Are Wrong: 17/23, 29/43 and 31/37 do not yield 2714/5074 when multiplied by any common integer factor. For example, 17/23 would correspond to 17k/23k, which cannot match 2714/5074 for an integer k. Hence those options are not equivalent to the given fraction. Common Pitfalls: Students may stop after dividing by a small factor like 2 and assume the result is in simplest form. However, it is important to check if a larger common factor exists. Using the GCD or the Euclidean algorithm helps avoid premature stopping and ensures the fraction is fully reduced. Final Answer: The fraction 2714/5074 in lowest terms is 23/43.

Question 8

The sum of twice a fraction and three times its reciprocal is 29/3. What is the value of this fraction?

Accepted Answer

Introduction / Context: This question again explores algebraic relationships between a fraction and its reciprocal. The expression given combines twice the fraction and three times its reciprocal to produce 29/3. Such problems test your skill in forming and solving quadratic equations based on verbal descriptions. Given Data / Assumptions: Let the fraction be x, with x ≠ 0. Its reciprocal is 1/x. The given condition is 2x + 3 * (1/x) = 29/3. We must find the value of x from the options. Concept / Approach: We transform the verbal relation into an algebraic equation in x. To get rid of the denominator x, we multiply throughout by x. This produces a quadratic equation, which we solve using the quadratic formula. Then we compare the obtained roots to the answer choices and pick the appropriate one. Step-by-Step Solution: Step 1: Start from the equation 2x + 3/x = 29/3. Step 2: Multiply both sides by 3x to eliminate the denominator: 3x * 2x + 3x * (3/x) = 3x * 29/3. Step 3: Simplify: 6x^2 + 9 = 29x. Step 4: Rearrange to quadratic form: 6x^2 - 29x + 9 = 0. Step 5: Use the quadratic formula: x = [29 ± sqrt(29^2 - 4 * 6 * 9)] / (2 * 6). Step 6: Compute the discriminant: 29^2 = 841; 4 * 6 * 9 = 216; so discriminant = 841 - 216 = 625. Step 7: The square root of 625 is 25, so x = (29 ± 25) / 12. Step 8: The roots are x = (29 + 25)/12 = 54/12 = 9/2 and x = (29 - 25)/12 = 4/12 = 1/3. Verification / Alternative Check: Test x = 9/2. Then the reciprocal is 2/9. Compute 2x + 3/x = 2 * (9/2) + 3 * (2/9) = 9 + 6/9 = 9 + 2/3. Convert 9 to thirds: 9 = 27/3, so 27/3 + 2/3 = 29/3, matching the given value. The other root 1/3 also satisfies the equation but does not appear in the options provided. Therefore, 9/2 is the correct choice. Why Other Options Are Wrong: 2/9, 5/4 and 4/5 do not satisfy the equation when substituted into 2x + 3/x. In each case, evaluating the expression will give a value different from 29/3, so these options must be rejected. Common Pitfalls: Learners sometimes mis-handle the multiplication by 3x and either forget a term or misplace coefficients. Another common issue is an error in computing the discriminant for the quadratic formula. Writing each intermediate step clearly helps reduce such mistakes. Final Answer: The fraction that satisfies the condition is 9/2.

Question 9

The sum of four times a fraction and six times its reciprocal is 11. What is the value of this fraction?

Accepted Answer

Introduction / Context: Yet again, we are given a relationship between a fraction and its reciprocal, this time involving four times the fraction and six times its reciprocal. This type of problem is standard in algebra for competitive exams and reinforces the technique of forming and solving quadratic equations. Given Data / Assumptions: Let the fraction be x, with x ≠ 0. Its reciprocal is 1/x. The equation is 4x + 6 * (1/x) = 11. We must find x among the given options. Concept / Approach: We translate the verbal condition into algebra, multiply to clear the denominator, and solve the resulting quadratic equation. After finding the roots, we choose the root that appears in the given choices and fits the idea of a reasonable fraction in this context. Step-by-Step Solution: Step 1: Begin with the equation 4x + 6/x = 11. Step 2: Multiply both sides by x to eliminate the denominator: x * 4x + x * (6/x) = 11x. Step 3: Simplify: 4x^2 + 6 = 11x. Step 4: Rearrange to standard form: 4x^2 - 11x + 6 = 0. Step 5: Try to factor the quadratic: we look for two numbers whose product is 4 * 6 = 24 and whose sum is -11. Step 6: The pair (-3) and (-8) multiplies to 24 and adds to -11. So factor as (4x - 3)(x - 2) = 0. Step 7: Set each factor equal to zero: 4x - 3 = 0 or x - 2 = 0. Step 8: Solve: from 4x - 3 = 0, we get x = 3/4; from x - 2 = 0, we get x = 2. Verification / Alternative Check: Check x = 3/4 in the original equation. The reciprocal is 4/3. Compute 4x + 6/x = 4 * (3/4) + 6 * (4/3). First, 4 * (3/4) = 3. Next, 6 * (4/3) = 24/3 = 8. Sum: 3 + 8 = 11, which matches the condition. The other root x = 2 also satisfies the equation but is not among the options, and the problem clearly focuses on fractional forms. Therefore, 3/4 is the correct answer from the given choices. Why Other Options Are Wrong: 4/3, 4/7 and 7/4 produce different values when substituted into 4x + 6/x. None of them give exactly 11, so they do not satisfy the equation and must be discarded. Common Pitfalls: Some students attempt to cross-multiply incorrectly or make errors in factorization. Another mistake is to forget that a quadratic can yield two roots and then failing to verify which root matches the answer set. For accuracy, always test each candidate solution in the original equation. Final Answer: The required fraction is 3/4.

Question 10

Convert the decimal number 0.0068 into a vulgar fraction and express your answer in lowest terms.

Accepted Answer

Introduction / Context: Converting terminating decimals into vulgar fractions is a basic skill in arithmetic. A vulgar fraction is simply an ordinary fraction like a/b. In this problem, the decimal 0.0068 must be written as a fraction and simplified to lowest terms. This type of conversion is often used in problems involving precision, measurement and ratios. Given Data / Assumptions: Decimal given: 0.0068. We assume it is exact and terminating. We must express it as a fraction in lowest terms. Concept / Approach: To convert a terminating decimal to a fraction: Count the number of digits after the decimal point. Write the decimal as an integer divided by a power of 10 (10, 100, 1000, etc.). Simplify the resulting fraction by cancelling common factors between numerator and denominator. Here, 0.0068 has four digits after the decimal point, so we will use 10^4 = 10000 as the denominator initially. Step-by-Step Solution: Step 1: Recognize that 0.0068 has four decimal places. Step 2: Write 0.0068 as 68/10000, because moving the decimal point four places to the right gives 68. Step 3: Now simplify the fraction 68/10000. Step 4: Find the greatest common divisor of 68 and 10000. Both are divisible by 4. Step 5: Divide the numerator by 4: 68 ÷ 4 = 17. Step 6: Divide the denominator by 4: 10000 ÷ 4 = 2500. Step 7: So 68/10000 simplifies to 17/2500. Step 8: Check if 17 and 2500 have any further common factors. 17 is prime and does not divide 2500, so 17/2500 is already in lowest terms. Verification / Alternative Check: Convert 17/2500 back to a decimal. Since 2500 = 25 * 100, we can write 17/2500 = (17/25) / 100. Now 17/25 = 0.68. Dividing 0.68 by 100 shifts the decimal two more places to the left, giving 0.0068. This confirms our fraction is correct. Why Other Options Are Wrong: 7/1250 ≈ 0.0056, which is less than 0.0068. 17/5000 ≈ 0.0034, which is half of 0.0068, so it is not correct. 7/1750 ≈ 0.004, also not equal to 0.0068. Only 17/2500 converts exactly to the given decimal. Common Pitfalls: Students sometimes miscount decimal places and write 0.0068 as 68/1000 instead of 68/10000. Another error is stopping simplification too early, for example leaving the answer as 68/10000. Always look for common divisors to reduce the fraction completely. Final Answer: The decimal 0.0068 expressed as a vulgar fraction in lowest terms is 17/2500.

Question 11

When 95.75 is multiplied by 0.2554, how many digits will appear to the right of the decimal point in the final product?

Accepted Answer

Introduction: This question examines your understanding of how decimal places behave during multiplication of decimal numbers. Instead of requiring full multiplication, the problem focuses only on counting the total number of digits to the right of the decimal point in the resulting product. Given Data / Assumptions: First number: 95.75. Second number: 0.2554. We must find the number of digits to the right of the decimal point in the product 95.75 * 0.2554. No rounding or truncation is applied; we consider the exact product. Concept / Approach: When two decimal numbers are multiplied, the total number of decimal places in the product equals the sum of the decimal places in each factor. We do not need to actually multiply the numbers to answer this question; we simply count decimal digits in each number and add them. Step-by-Step Solution: Step 1: Count decimal digits in 95.75. It has 2 digits after the decimal point (7 and 5). Step 2: Count decimal digits in 0.2554. It has 4 digits after the decimal point (2, 5, 5, and 4). Step 3: Total decimal digits in the product = 2 + 4 = 6. Step 4: Therefore, the product 95.75 * 0.2554 will have 6 digits to the right of the decimal point. Verification / Alternative check: If you remove the decimals temporarily, you can think of 95.75 as 9575 / 100 and 0.2554 as 2554 / 10000. Their product is (9575 * 2554) / (100 * 10000) = some integer divided by 10^6, which clearly has 6 decimal places. This confirms the result without needing the exact product value. Why Other Options Are Wrong: 7, 8, or 9: These would require more decimal digits than the sum of the original decimal places, which is not possible with straightforward multiplication. 5: This would require one fewer decimal place than the sum, which does not match the rule that decimal places add in multiplication. Common Pitfalls: Learners sometimes miscount decimal places, especially when there is a leading zero before the decimal. Remember that only digits after the decimal point are counted, and the total number in the product is the sum from both factors. There is no need to perform the full multiplication for such questions. Final Answer: The product will have 6 digits to the right of the decimal point.

Question 12

When the decimal 0.36 is written as a fraction in its simplest (lowest terms) form, what is the sum of the numerator and the denominator of that fraction?

Accepted Answer

Introduction / Context: This question checks your understanding of converting terminating decimals into fractions and then simplifying them to lowest terms. Such decimal-to-fraction conversions are common in aptitude tests and help reinforce the link between decimal notation and rational numbers. Given Data / Assumptions: Decimal given: 0.36. We must express 0.36 as a fraction in simplest form. Then we must find the sum of the numerator and the denominator. Concept / Approach: A terminating decimal can be written as a fraction with a power of 10 in the denominator and the decimal digits as the numerator, and then simplified by dividing numerator and denominator by their greatest common divisor (GCD). Here 0.36 has two decimal places, so we first write it as 36/100 and then reduce the fraction to lowest terms. Step-by-Step Solution: Step 1: Write 0.36 as a fraction with denominator 100: 0.36 = 36/100. Step 2: Find the greatest common divisor of 36 and 100. Step 3: The prime factors of 36 are 2 * 2 * 3 * 3, and the prime factors of 100 are 2 * 2 * 5 * 5. The common factors are 2 * 2 = 4. Step 4: Divide numerator and denominator by 4: 36 ÷ 4 = 9, and 100 ÷ 4 = 25. Step 5: So 0.36 = 36/100 = 9/25 in simplest form. Step 6: Sum of numerator and denominator = 9 + 25 = 34. Verification / Alternative check: Convert 9/25 back to a decimal: 9 ÷ 25 = 0.36 exactly, so 9/25 is indeed equivalent to 0.36 and is fully simplified because 9 and 25 have no common factor other than 1. Why Other Options Are Wrong: 15 might be obtained by incorrectly using 3/10 instead of 9/25 as the simplified fraction. 64 and 13 do not correspond to any correct pair of numerator and denominator representing 0.36 in lowest terms. Common Pitfalls: A frequent error is to stop at 36/100 without simplifying, or to divide by a smaller common factor like 2 or 5 instead of the full GCD 4. Another mistake is to miscalculate the GCD and obtain a fraction that still can be reduced further. Final Answer: The sum of the numerator and denominator is 34.

Question 13

A decimal number 47.2506 is expressed in terms of place value units as 47.2506 = 4*A + 7*B + 2*C + 5*D + 6*E, where A, B, C, D and E represent the standard place value units 10, 1, 0.1, 0.01 and 0.0001 respectively. What is the value of 5*A + 3*B + …

Accepted Answer

Introduction / Context: This question assesses your understanding of decimal place values and how a decimal number can be decomposed into a linear combination of place value units. By carefully matching the digits of the decimal 47.2506 with their corresponding place values, you can identify the numerical values of A, B, C, D and E. You then substitute these into another linear expression to find its value. Given Data / Assumptions: Number given: 47.2506. Representation: 47.2506 = 4*A + 7*B + 2*C + 5*D + 6*E. A, B, C, D and E represent standard decimal place value units. We are told that A = 10, B = 1, C = 0.1, D = 0.01, E = 0.0001. We must compute the value of 5*A + 3*B + 6*C + D + 3*E. Concept / Approach: The number 47.2506 has digits in the tens, ones, tenths, hundredths and ten-thousandths places. Therefore, 4 is in the tens place, 7 is in the ones place, 2 is in the tenths place, 5 is in the hundredths place, and 6 is in the ten-thousandths place. These correspond exactly to multiplying 4, 7, 2, 5 and 6 by 10, 1, 0.1, 0.01 and 0.0001 respectively. After confirming this mapping, we plug the same values of A, B, C, D and E into the new expression and evaluate. Step-by-Step Solution: Step 1: Confirm the mapping: A = 10, B = 1, C = 0.1, D = 0.01, E = 0.0001. Step 2: Check the given decomposition: 4*A = 4 * 10 = 40. Step 3: 7*B = 7 * 1 = 7, so 40 + 7 = 47. Step 4: 2*C = 2 * 0.1 = 0.2, bringing the total to 47.2. Step 5: 5*D = 5 * 0.01 = 0.05, making the running total 47.25. Step 6: 6*E = 6 * 0.0001 = 0.0006, giving 47.2506, which matches the original decimal. Step 7: Now evaluate 5*A + 3*B + 6*C + D + 3*E: 5*A = 5 * 10 = 50. Step 8: 3*B = 3 * 1 = 3, so subtotal is 50 + 3 = 53. Step 9: 6*C = 6 * 0.1 = 0.6, subtotal is 53.6. Step 10: D = 0.01, so subtotal becomes 53.61. Step 11: 3*E = 3 * 0.0001 = 0.0003, final result is 53.6103. Verification / Alternative check: You can quickly check whether any other interpretation of A, B, C, D, E would still satisfy the original equality 47.2506 = 4*A + 7*B + 2*C + 5*D + 6*E. Using the standard place values is the most natural and consistent interpretation and recreates the number exactly. Why Other Options Are Wrong: 55.6003, 53.603 and 153.6003 do not equal 5*A + 3*B + 6*C + D + 3*E when A, B, C, D and E are the standard decimal place values. Common Pitfalls: Learners may confuse which digit corresponds to which place value or may miscalculate small decimal contributions such as E = 0.0001. Rounding intermediate calculations too aggressively can also lead to slight but significant errors in the final result. Final Answer: The value of the expression 5*A + 3*B + 6*C + D + 3*E is 53.6103.

Question 14

If 1.5x = 0.04y for two positive real numbers x and y, what is the value of the expression (y - x) / (y + x)?

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Introduction / Context:This question tests your ability to manipulate linear equations in two variables and then convert that relationship into a ratio expression. Such algebraic manipulation appears frequently in aptitude exams, especially where variables are related by decimals, and you must express one in terms of the other and then substitute into a new expression. Given Data / Assumptions: We are given the equation 1.5x = 0.04y. x and y are positive real numbers. We must find the value of (y - x) / (y + x). No specific numerical values of x and y are provided, so the expression must simplify to a constant. Concept / Approach:The key idea is to express y in terms of x, or x in terms of y, using the given linear equation. Once we know the ratio y / x, we can rewrite (y - x) / (y + x) in terms of that ratio alone. This removes the need to know individual values and turns the problem into pure algebraic simplification. Step-by-Step Solution:Start from the given equation: 1.5x = 0.04y.Rearrange to get y in terms of x: y = (1.5 / 0.04) * x.Compute the fraction 1.5 / 0.04 = 150 / 4 = 37.5, so y = 37.5x.Now form the required expression: (y - x) / (y + x) = (37.5x - x) / (37.5x + x).Simplify numerator and denominator: (36.5x) / (38.5x).Cancel x (since x is positive and nonzero): 36.5 / 38.5.Write these as fractions: 36.5 = 365 / 10 and 38.5 = 385 / 10.So the ratio becomes (365 / 10) / (385 / 10) = 365 / 385.Divide numerator and denominator by 5 to get 73 / 77. Verification / Alternative check:We can verify by choosing any convenient positive value for x and computing y from the relation. For example, let x = 2. Then y = 37.5 * 2 = 75. Now calculate (y - x) / (y + x) = (75 - 2) / (75 + 2) = 73 / 77. This matches the simplified algebraic result, confirming our answer is consistent and independent of the particular value of x chosen. Why Other Options Are Wrong:Option 71/77 is close but would come from a different ratio, and does not match the derived algebraic simplification. Option 36/37 would arise if the constants were different and is too large compared to the true ratio. Option 3/5 is a much simpler fraction and does not come from any correct manipulation of 1.5x = 0.04y. Option 37/73 is an inverted and slightly modified form and is not supported by any valid transformation of the given relation. Common Pitfalls:A common mistake is mishandling the decimal division 1.5 / 0.04, for example treating it as 1.5 / 4 or 0.15 / 0.04. Another error is substituting incorrectly into (y - x) / (y + x), such as forgetting to factor out x or cancel it. Some students also attempt to assign random values to x and y that do not satisfy the original equation, leading to inconsistent answers. Finally, forgetting to simplify the final fraction fully can cause you to miss the reduced form 73 / 77. Final Answer:The value of the expression (y - x) / (y + x), given that 1.5x = 0.04y, is 73/77.

Question 15

Evaluate the product 3 × 0.3 × 0.03 × 0.003 × 30 and choose the correct decimal value.

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Introduction / Context:This problem checks your fluency with multiplication of decimals and understanding of how place values shift when you multiply several decimal numbers together. Such questions appear frequently in decimal fractions and number system sections of aptitude tests and are designed to test both calculation accuracy and place value concepts. Given Data / Assumptions: The numbers to be multiplied are 3, 0.3, 0.03, 0.003, and 30. All values are real numbers, and the operations are simple multiplication. We have to compute the exact decimal product and match it with one of the given options. No rounding is required because the options are given with precise decimal digits. Concept / Approach:The key concept is that multiplying by 10, 100, and so on shifts the decimal point to the right, while multiplying by tenths, hundredths, or thousandths shifts it to the left. A helpful strategy is to separate the calculation into a product of whole numbers and a power of 10 derived from counting decimal places. Alternatively, you can multiply step by step while being very careful with decimal positions. Step-by-Step Solution:First group the whole number factors: 3 × 30 = 90.Next, multiply the decimal factors: 0.3 × 0.03 × 0.003.Write them as powers of 10: 0.3 = 3 × 10^-1, 0.03 = 3 × 10^-2, 0.003 = 3 × 10^-3.Then 0.3 × 0.03 × 0.003 = (3 × 3 × 3) × 10^(-1-2-3) = 27 × 10^-6.Now multiply this with 90: 90 × 27 × 10^-6.Compute 90 × 27 = 2430.So the overall product is 2430 × 10^-6 = 0.00243.Therefore, the required value is 0.00243. Verification / Alternative check:As a quick check, you can multiply directly on a calculator or carefully by hand: 3 × 0.3 = 0.9, 0.9 × 0.03 = 0.027, 0.027 × 0.003 = 0.000081, and 0.000081 × 30 = 0.00243. Both methods produce the same answer, confirming that our place value handling is correct and that no decimal has been misplaced. Why Other Options Are Wrong:Option 0.0000243 is smaller by a factor of 100 and would result from shifting the decimal two places too far to the left. Option 0.000243 is smaller by a factor of 10, indicating one extra left shift. Option 0.0243 is larger by a factor of 10, suggesting one fewer left shift. Option 0.00000243 is off by a factor of 1000 and reflects an even more serious mistake with powers of 10. Common Pitfalls:Learners often miscount the total number of decimal places and shift the decimal incorrectly. Another common mistake is to ignore that both 3 and 30 are whole numbers, so they do not contribute additional decimal places. Some may multiply 0.3, 0.03, and 0.003 incorrectly due to confusion about tenths, hundredths, and thousandths. Careful writing of intermediate steps and use of scientific notation are effective ways to avoid these errors. Final Answer:The value of 3 × 0.3 × 0.03 × 0.003 × 30 is 0.00243.

Question 16

When the recurring decimal 0.232323... is converted into a fraction in simplest form, what is the resulting fraction?

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Introduction / Context:This question examines your understanding of recurring decimals and how to convert them into rational numbers in fractional form. Knowing how to express repeating decimals as fractions is important in number system topics and helps connect decimal representation with exact fractional values used in algebra and higher mathematics. Given Data / Assumptions: The given decimal is 0.232323..., where the block 23 repeats indefinitely. We assume the repetition continues forever without stopping. We must find a fraction in lowest terms that is exactly equal to this recurring decimal. No approximation or rounding is allowed; the answer must be exact. Concept / Approach:The standard method for converting a recurring decimal into a fraction uses algebraic manipulation. We assign the repeating decimal to a variable, multiply by a power of 10 that shifts the repeating block to the left of the decimal point, and then subtract the original expression. This cancellation removes the repeating part and yields a simple linear equation in one variable. Step-by-Step Solution:Let x = 0.232323... where 23 repeats indefinitely.Since two digits repeat, multiply x by 100: 100x = 23.232323...Now subtract the original x from this: 100x - x = 23.232323... - 0.232323...This gives 99x = 23.Solve for x: x = 23 / 99.Check if 23 / 99 is already in simplest form. The numerator 23 is a prime number and does not divide 99, so the fraction is already reduced.Therefore, 0.232323... = 23 / 99 in simplest fractional form. Verification / Alternative check:We can verify by performing long division of 23 by 99. The quotient begins 0.232323..., and the block 23 will repeat indefinitely. This confirms that 23 / 99 is indeed the exact representation of the recurring decimal. Using a calculator in repeating mode or manual division both support this conclusion and show that our algebraic method is reliable. Why Other Options Are Wrong:Option 1/5 equals 0.2, which does not have a repeating block 23. Option 2/9 equals 0.222..., so the repeating block is 2, not 23. Option 23/100 equals 0.23 exactly, with no repetition, so it is a terminating decimal. Option 21/90 simplifies to 7/30, which equals 0.2333..., again not matching 0.232323... . None of these capture the specific two digit repeating pattern that appears in the given decimal. Common Pitfalls:Students sometimes multiply by 10 instead of 100, which fails to align the repeating block properly. Another mistake is to forget to subtract the original x, leaving the repeating part in place. Some learners also mistakenly round 0.232323... to 0.23, treating it as a terminating decimal, which leads to 23/100 and an incorrect answer. Careful attention to the length of the repeating block and disciplined use of algebra avoid these issues. Final Answer:The recurring decimal 0.232323... is exactly equal to the fraction 23/99 in simplest form.

Question 17

In one hour there are 3600 seconds. What decimal part of an hour is represented by a single second? Give the answer correct to five decimal places.

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Introduction / Context:This question connects time measurement with decimal fractions. It asks you to express one second as a decimal part of an hour, which is common in speed, work, and time calculations where hours are used as the main unit and smaller time units must be converted accurately. Given Data / Assumptions: 1 hour = 60 minutes. 1 minute = 60 seconds. Therefore, 1 hour = 60 × 60 = 3600 seconds. We wish to express 1 second as a fraction, and then as a decimal, of 1 hour. The answer choices are in decimal form and are close in value, so attention to place value and rounding is important. Concept / Approach:We interpret the phrase "what decimal of an hour" as the ratio (time in seconds) / (3600 seconds). Since we want the decimal for one second, we compute 1 divided by 3600. The resulting value is then rounded to the required number of decimal places to match the pattern of the options. This type of ratio-to-decimal conversion is standard in decimal fraction problems. Step-by-Step Solution:Total seconds in one hour = 60 × 60 = 3600 seconds.We want the fraction of an hour represented by 1 second: this is 1 / 3600.Compute 1 / 3600 as a decimal: 1 ÷ 3600 ≈ 0.0002777777...To match the style of the options, round this to five decimal places.0.0002777777... rounds to 0.00028 (since the sixth decimal digit is 7, which causes rounding up).Therefore, one second is approximately 0.00028 of an hour. Verification / Alternative check:We can reverse check by multiplying 0.00028 by 3600. Compute 0.00028 × 3600 = 1.008. This is very close to 1 second, with the small difference due to rounding. If we used more decimal places, the product would be even closer to exactly 1. This confirms that 0.00028 is a valid rounded representation of one second as a decimal part of an hour. Why Other Options Are Wrong:Option 0.00027 is slightly smaller and would give 0.972 seconds when multiplied by 3600, which underestimates a second. Option 0.00026 and 0.00025 are even smaller and deviate more from the true value 1/3600. Option 0.00030 is larger and would correspond to 1.08 seconds of an hour, which overestimates the actual time. Therefore, only 0.00028 is a reasonable rounded match to 1 / 3600. Common Pitfalls:Some learners mistakenly divide 1 by 60 instead of by 3600, effectively computing the fraction of a minute instead of an hour. Others misplace the decimal when dividing by large numbers, producing 0.0028 or similar values that are ten times or one hundred times larger than the correct answer. Remembering that an hour contains 3600 seconds and performing the division carefully avoids these errors. Final Answer:One second is approximately 0.00028 of an hour when expressed as a decimal fraction.

Question 18

Which of the following fractions has a value greater than 3/4 but less than 5/6?

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Introduction / Context:This problem checks your understanding of comparing fractions and locating them on the number line. Rather than converting every fraction into a decimal with a calculator, the question encourages you to use basic fraction comparison techniques and reasoning about sizes of fractions relative to familiar benchmarks like three quarters and five sixths. Given Data / Assumptions: We are given two bounds: 3/4 and 5/6. We must identify which option lies strictly between these two fractions. The options are 3/5, 4/5, 7/5, 9/5, and 5/7. All fractions are positive rational numbers. Concept / Approach:There are two main ways to compare fractions: either convert to decimals or bring them to a common denominator. Here, 3/4 and 5/6 already share small denominators, so we can convert to decimals quickly or find approximate values. We also note that fractions greater than 1 cannot lie between 3/4 and 5/6, so we can eliminate such options immediately. Step-by-Step Solution:First, compute approximate decimal values of the bounds.3/4 = 0.75.5/6 ≈ 0.8333.Now consider each option and compare.3/5 = 0.6, which is less than 0.75, so it is too small.4/5 = 0.8, which satisfies 0.75 < 0.8 < 0.8333.7/5 = 1.4 and 9/5 = 1.8, which are both greater than 1, so they cannot lie between 3/4 and 5/6.5/7 ≈ 0.714, less than 0.75, also too small.Thus, only 4/5 lies strictly between 3/4 and 5/6. Verification / Alternative check:Alternatively, we can compare 4/5 without decimals by using cross multiplication. To check if 3/4 < 4/5, compare 3 × 5 and 4 × 4. We get 15 and 16, and since 15 < 16, 3/4 < 4/5. To check if 4/5 < 5/6, compare 4 × 6 and 5 × 5. We get 24 and 25, and since 24 < 25, 4/5 < 5/6. Both inequalities hold, confirming that 4/5 is indeed between the two given fractions. Why Other Options Are Wrong:3/5 and 5/7 are both less than 3/4 and therefore cannot satisfy the condition. 7/5 and 9/5 exceed 1, which is much larger than 5/6, so they are automatically excluded. None of these fractions meet the requirement of being simultaneously greater than 3/4 and less than 5/6, leaving 4/5 as the only valid choice. Common Pitfalls:Some students misinterpret the phrase greater than 3/4 and less than 5/6 and only check one of the two inequalities. Others compare fractions by looking only at numerators or denominators without considering the full fraction. Forgetting that values greater than 1 cannot sit between two proper fractions less than 1 is another frequent error. Consistent use of cross multiplication or decimal conversion helps prevent these mistakes. Final Answer:The fraction that lies between 3/4 and 5/6 is 4/5.

Question 19

A tailor has 37.5 metres of cloth. He uses each metre of cloth to make 8 equal pieces. How many such pieces in total can he make from the entire length of cloth?

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Introduction / Context:This is a straightforward application of the unitary method and decimal multiplication. The question checks whether you can interpret the statement correctly and multiply a decimal quantity by a whole number to get the total count of pieces formed from a given length of cloth. Given Data / Assumptions: Total cloth available = 37.5 metres. From each metre of cloth, the tailor can cut 8 equal pieces. The cutting process is assumed to be exact with no wastage. We must find the total number of equal pieces that can be made. Concept / Approach:The key idea is to treat the number of pieces per metre as a constant rate, then multiply that rate by the total number of metres. This is a direct use of the unitary method: if 1 metre gives 8 pieces, then 37.5 metres give 37.5 times 8 pieces. Handling the decimal 37.5 correctly is essential for obtaining the right answer. Step-by-Step Solution:Pieces obtained from 1 metre of cloth = 8 pieces.Total length of cloth = 37.5 metres.Total pieces = 37.5 × 8.Compute 37.5 × 8 by breaking it: 30 × 8 = 240 and 7.5 × 8 = 60.Add the partial products: 240 + 60 = 300.Therefore, the tailor can make 300 pieces from 37.5 metres of cloth. Verification / Alternative check:We can verify using fraction representation. Write 37.5 as 75/2. Then total pieces = (75/2) × 8 = 75 × 4 = 300. This confirms that both decimal multiplication and fractional multiplication lead to the same total count, so our calculation is consistent and correct. Why Other Options Are Wrong:Option 360 would correspond to 45 metres of cloth at 8 pieces per metre, which is more cloth than given. Option 400 would require 50 metres, and option 450 would require 56.25 metres. Option 280 would correspond to 35 metres of cloth. None of these match the given 37.5 metres, so they are not valid results for this scenario. Common Pitfalls:A frequent mistake is misreading 37.5 as 37 or 35, which leads to incorrect multiplication. Another error is mishandling the decimal arithmetic, for example treating 37.5 × 8 as 37.5 × 4 or forgetting part of the calculation. Breaking the product into simpler parts or converting the decimal to a fraction helps avoid such slips and ensures accurate results. Final Answer:The tailor can make a total of 300 equal pieces from 37.5 metres of cloth.

Question 20

Which of the following rewritten products is exactly equal to 54.327 × 357.2 × 0.0057?

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Introduction / Context:This question tests your ability to manipulate decimal numbers by shifting decimal points and compensating with powers of 10. It is a typical decimal fraction question where you must recognize that moving the decimal point changes the magnitude of a number and that equivalent products can be formed by balancing those shifts across all factors. Given Data / Assumptions: Original product: 54.327 × 357.2 × 0.0057. Several alternative products are provided, each formed by moving decimal points. We must find which alternative product has the same numerical value as the original expression. No approximation is allowed; the equality must be exact. Concept / Approach:When you move the decimal point to the left, you are effectively dividing the number by 10 for each place shifted. Moving it to the right multiplies it by 10. If you change one factor by a power of 10, you must adjust another factor by an opposite power of 10 to keep the overall product unchanged. The core idea is to track how many decimal places have been shifted and translate that into powers of 10. Step-by-Step Solution:Start with 54.327. Moving the decimal one place left gives 5.4327, which is 54.327 ÷ 10.Next consider 357.2. Moving the decimal two places left gives 3.572, which is 357.2 ÷ 100.Together, these two changes divide the original product by 10 × 100 = 1000.To keep the product unchanged, we must multiply the third factor 0.0057 by 1000.0.0057 × 1000 = 5.7.So an equivalent product is 5.4327 × 3.572 × 5.7.Therefore, option 5.4327 × 3.572 × 5.7 is exactly equal to the original expression. Verification / Alternative check:We can verify numerically. Compute the original product and the candidate product. Both 54.327 × 357.2 × 0.0057 and 5.4327 × 3.572 × 5.7 evaluate to the same decimal value when calculated carefully. This confirms that our reasoning with powers of 10 and decimal shifts is accurate. Why Other Options Are Wrong:Option 5.4327 × 3.572 × 0.57 only multiplies the last factor by 100, while the first two factors together divided the product by 1000, so the overall product becomes one tenth of the original. Option 54327 × 3572 × 0.0000057 changes factors by far larger powers of 10 in an unbalanced way, not preserving equality. Option 54.327 × 3.572 × 0.57 changes only some factors and again does not balance the net power of 10. Therefore, these are not numerically equal to the original product. Common Pitfalls:Students sometimes move decimal points without explicitly thinking in terms of powers of 10, which makes it easy to lose track of how much the product has changed. Another common mistake is to assume that because numbers look similar, the expressions must be equal, without checking the net number of decimal shifts. Always count the total power of 10 applied across all factors and ensure that the net effect is zero if you want an equivalent product. Final Answer:The expression equivalent to 54.327 × 357.2 × 0.0057 is 5.4327 × 3.572 × 5.7.

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