Q: A person was asked to state his age in years. He replied: Take my age 3 years from now, multiply it by 3, then subtract 3 times my age 3 years ago, and the result will equal my present age. How old is this person now?

Introduction / Context: This is a classic age puzzle that tests your ability to translate a verbal description into an algebraic equation. Once the equation is formed, solving it is straightforward. Such puzzles frequently appear in aptitude exams to evaluate logical thinking and algebra skills. Given Data / Assumptions: Let the present age of the person be A years. Age 3 years from now is A + 3. Age 3 years ago was A - 3. The statement says: 3 × (age 3 years from now) minus 3 × (age 3 years ago) equals the present age A. Concept / Approach: We convert the language into an algebraic equation. Then we simplify the equation and solve for A. As the numbers are small and linear, it is a simple equation in one variable. Step-by-Step Solution: Step 1: Assume present age = A years. Step 2: Age 3 years from now = A + 3. Step 3: Age 3 years ago = A - 3. Step 4: According to the condition: 3 × (A + 3) - 3 × (A - 3) = A. Step 5: Expand the left side: 3A + 9 - (3A - 9) = A. Step 6: Simplify: 3A + 9 - 3A + 9 = A, so 18 = A. Step 7: Therefore, the present age of the person is 18 years. Verification / Alternative check: Check the condition with A = 18. Age 3 years from now: 21. Age 3 years ago: 15. 3 × 21 = 63 and 3 × 15 = 45. The difference is 63 - 45 = 18, which equals the present age. This confirms the solution. Why Other Options Are Wrong: 20: If A = 20, the calculation would give 3 × 23 - 3 × 17 = 69 - 51 = 18, not 20. 24: If A = 24, 3 × 27 - 3 × 21 = 81 - 63 = 18, not 24. 30: If A = 30, 3 × 33 - 3 × 27 = 99 - 81 = 18, not 30. 22: Similarly gives a result different from 22. Only 18 satisfies the equation. Common Pitfalls: The main difficulty is correctly interpreting the verbal description. Some learners may forget the negative sign when subtracting 3 × (A - 3) or may misread the phrase "3 years ago." Writing each step of the equation explicitly and expanding carefully prevents sign errors. Final Answer: The person is currently 18 years old.

Question 1

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

Accepted Answer

Introduction / Context: Decimals and fractions are two different ways of representing the same rational number. Competitive exams often test your ability to convert between these forms and then perform small operations like adding numerator and denominator. Here we work with the decimal 0.36 and convert it into a fraction in simplest form, then find the sum of its numerator and denominator. Given Data / Assumptions: The given decimal is 0.36. We must express 0.36 as a fraction in simplest (lowest) terms. After simplification, we must add the numerator and the denominator. Concept / Approach: A terminating decimal can be converted into a fraction by writing it over a power of 10 and simplifying. For two decimal places, we use 100 as the denominator. Then we reduce the fraction by dividing numerator and denominator by their greatest common divisor (GCD). Finally, we add numerator and denominator to get the required sum. Step-by-Step Solution: Step 1: Write 0.36 as a fraction: 0.36 = 36 / 100. Step 2: Find common factors of 36 and 100. The GCD of 36 and 100 is 4. Step 3: Divide numerator and denominator by 4: 36 / 4 = 9 and 100 / 4 = 25. Step 4: So, 0.36 in simplest form is 9 / 25. Step 5: Now add the numerator and denominator: 9 + 25 = 34. Verification / Alternative check: We can convert 9 / 25 back to a decimal to confirm. Divide 9 by 25: 9 ÷ 25 = 0.36. Since this matches the original decimal, the fraction 9 / 25 is correct. The sum 9 + 25 = 34 is therefore reliable. Why Other Options Are Wrong: 15: This would correspond to some incorrect simplified fraction such as 7 / 8 (7 + 8 = 15), which does not equal 0.36. 64: This would suggest a very different numerator and denominator, not matching 0.36. 13: This is too small to be the sum of numerator and denominator of 0.36 in simplest form. 49: Also does not relate to 9 / 25 and is just a distractor value. Common Pitfalls: A common mistake is to forget to simplify 36 / 100 and directly add 36 + 100 = 136, which is wrong. Another error is to reduce the fraction incorrectly by dividing by a number that is not the true GCD, leading to a non-simplest form. Always check divisibility carefully. Final Answer: The simplified fraction is 9 / 25 and the required sum of numerator and denominator is 34.

Question 2

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

Accepted Answer

Introduction / Context: This question tests algebraic manipulation and understanding of ratios between two variables. We are given a linear relationship between x and y and asked to evaluate the ratio (y - x) / (y + x) without knowing the actual values of x and y. Such questions appear frequently in aptitude tests to check symbolic reasoning skills. Given Data / Assumptions: Relation between variables: 1.5x = 0.04y. We need to compute the value of (y - x) / (y + x). x and y are nonzero real numbers to keep the expressions meaningful. Concept / Approach: From the equation 1.5x = 0.04y, we can find the ratio y / x. Once that ratio is known, we express y in terms of x and substitute into (y - x) / (y + x). The variable x cancels out, leaving a pure numerical fraction that can be simplified. Step-by-Step Solution: Step 1: Start with 1.5x = 0.04y. Step 2: Rearrange to get y / x = 1.5 / 0.04. Step 3: Compute 1.5 / 0.04 = 150 / 4 = 37.5, so y = 37.5x. Step 4: Substitute into (y - x) / (y + x): (37.5x - x) / (37.5x + x). Step 5: Simplify numerator and denominator: (36.5x) / (38.5x). Step 6: Cancel x: (y - x) / (y + x) = 36.5 / 38.5. Step 7: Multiply top and bottom by 10: 36.5 / 38.5 = 365 / 385. Step 8: Divide both numerator and denominator by 5: 365 / 385 = 73 / 77. Verification / Alternative check: Take a convenient value of x, for example x = 2. Then y = 37.5 × 2 = 75. Now compute (y - x) / (y + x) = (75 - 2) / (75 + 2) = 73 / 77. This matches the derived fraction, confirming the result. Why Other Options Are Wrong: 7.3/77: This incorrectly places the decimal in the numerator and does not arise from correct simplification. 730/77 and 7300/77: These are much larger ratios and would require y to be enormously bigger relative to x, which is not implied by the equation. 7/77: This is a very small ratio and not consistent with the correct derivation. Common Pitfalls: Mistakes often occur when dividing decimals, such as mishandling 1.5 / 0.04. Forgetting to cancel x in numerator and denominator can also lead to unnecessarily complex expressions. Carefully converting decimals to fractions helps avoid these errors. Final Answer: The value of the expression (y - x) / (y + x) is 73/77.

Question 3

Calculate the exact value of the product 3 × 0.3 × 0.03 × 0.003 × 30, expressed as a decimal number.

Accepted Answer

Introduction / Context: This problem checks your understanding of multiplying decimal numbers and correctly placing the decimal point in the final answer. When several decimals are multiplied, it is easy to miscount the total number of decimal places. A systematic approach avoids such mistakes and leads to the right product. Given Data / Assumptions: We must compute the product: 3 × 0.3 × 0.03 × 0.003 × 30. The result must be written as a decimal. No rounding is required; we want the exact value. Concept / Approach: One way is to multiply step by step while paying attention to decimal places. Another way is to write each decimal as an integer divided by a power of 10, then combine the powers. Both methods will yield the same result. We will use stepwise multiplication for clarity and then cross-check with the decimal-place method. Step-by-Step Solution: Step 1: Multiply 3 × 0.3 = 0.9. Step 2: Multiply the result by 0.03: 0.9 × 0.03 = 0.027. Step 3: Multiply by 0.003: 0.027 × 0.003 = 0.000081. Step 4: Finally, multiply by 30: 0.000081 × 30 = 0.00243. Step 5: Therefore, the exact product is 0.00243. Verification / Alternative check: Write each factor as a fraction: 3 = 3, 0.3 = 3 / 10, 0.03 = 3 / 100, 0.003 = 3 / 1000, and 30 = 30. The product of integers is 3 × 3 × 3 × 3 × 30 = 2430. The product of denominators is 10 × 100 × 1000 = 1000000. So the product is 2430 / 1000000 = 0.00243, confirming the result. Why Other Options Are Wrong: 0.0000243 and 0.000243: These are smaller by factors of 10 or 100, caused by misplacing the decimal point. 0.0243: This is larger by a factor of 10, also due to decimal miscounting. 0.243: This is larger by a factor of 100, clearly not matching the careful calculation. Common Pitfalls: Typical mistakes include multiplying the numbers but not correctly tracking the total number of decimal places, or rounding too early. Writing numbers as fractions over powers of 10 gives a clear view of how many zeros appear in the denominator and prevents such errors. Final Answer: The exact value of the product is 0.00243.

Question 4

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

Accepted Answer

Introduction / Context: Repeating or recurring decimals can always be expressed as rational numbers, that is, as fractions of integers. This question asks you to convert the recurring decimal 0.232323... (with 23 repeating) into a fraction in simplest form. Understanding this conversion is important for aptitude tests and basic algebra. Given Data / Assumptions: The decimal is 0.232323..., where the block "23" repeats indefinitely. We must express this as a fraction in lowest terms. No rounding is involved because the pattern is perfectly repeating. Concept / Approach: For a recurring decimal with two repeating digits, like 0.abab..., we can use a standard algebraic trick. Let x be the repeating decimal. Multiplying x by 100 shifts the decimal two places to the right. Subtracting x then cancels the repeating part, leaving a simple linear equation that we can solve for x as a fraction. Step-by-Step Solution: Step 1: Let x = 0.232323... Step 2: Multiply both sides by 100 to shift two decimal places: 100x = 23.232323... Step 3: Subtract the original x: 100x - x = 23.232323... - 0.232323... Step 4: The repeating parts cancel, giving 99x = 23. Step 5: Solve for x: x = 23 / 99. Step 6: Check for further simplification. Since 23 is a prime number and does not divide 99, the fraction is already in simplest form. Verification / Alternative check: Compute 23 / 99 as a decimal by long division. You will get 0.232323..., confirming that we have the correct fraction. The repeating block "23" appears continuously, matching the original decimal representation. Why Other Options Are Wrong: 1/5 = 0.2, which does not repeat as 0.232323.... 2/9 ≈ 0.222..., repeating only the digit 2, not "23". 23/100 = 0.23, which terminates and does not continue repeating. 3/10 = 0.3, again a terminating decimal, not a recurring one with block "23". Common Pitfalls: Learners sometimes confuse terminating and recurring decimals or incorrectly multiply by 10 instead of 100 when two digits repeat. It is crucial to match the number of zeros in the multiplier to the number of digits in the repeating block. Also, always check whether the fraction can be simplified further by checking common factors. Final Answer: The recurring decimal 0.232323... is equal to the fraction 23/99 in simplest form.

Question 5

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

Accepted Answer

Introduction / Context: Number line comparison questions test your understanding of the relative sizes of fractions. Here you must identify which option lies strictly between 3/4 and 5/6. This is an important skill in fractions, inequalities, and ordering of rational numbers. Given Data / Assumptions: Lower bound fraction: 3/4. Upper bound fraction: 5/6. Candidate fractions: 3/5, 4/5, 7/5, 9/5, and 2/3. We need a fraction greater than 3/4 and less than 5/6. Concept / Approach: To compare fractions, we can convert them to decimals or bring them to a common denominator. Since 3/4 and 5/6 are standard fractions, using decimal equivalents is quick: 3/4 = 0.75 and 5/6 ≈ 0.8333. Then we check each option to see which value lies between these two decimal numbers. Step-by-Step Solution: Step 1: Convert bounds to decimals. 3/4 = 0.75. Step 2: Compute 5/6 ≈ 0.8333. Step 3: Evaluate options: 3/5 = 0.6 (less than 0.75). 4/5 = 0.8 (between 0.75 and 0.8333). 7/5 = 1.4 (greater than 0.8333). 9/5 = 1.8 (greater than 0.8333). 2/3 ≈ 0.6667 (less than 0.75). Step 4: Only 4/5 has a value between 0.75 and 0.8333. Verification / Alternative check: Use common denominators. The least common multiple of 4 and 6 is 12. Then 3/4 = 9/12 and 5/6 = 10/12. Now, 4/5 written with denominator 60 is 48/60, which corresponds to 9.6/12. So 9/12 < 9.6/12 < 10/12, confirming that 4/5 lies between 3/4 and 5/6. Why Other Options Are Wrong: 3/5 and 2/3 are both less than 3/4, so they are below the lower bound. 7/5 and 9/5 are both greater than 1, so they lie above 5/6 and are not between the bounds. Common Pitfalls: A frequent mistake is to compare only numerators or denominators without converting properly, which can be misleading. Another error is to assume that a larger denominator always means a smaller fraction, which is not true if numerators differ. Always use decimal conversion or common denominators for reliable comparison. Final Answer: The only fraction that lies between 3/4 and 5/6 is 4/5.

Question 6

What is the rational number (fraction in simplest form) that is equivalent to the recurring decimal 0.125125125... ?

Accepted Answer

Introduction / Context: This problem involves converting a recurring decimal with a repeating block of three digits into an equivalent fraction. Understanding this method is useful whenever you see repeating patterns like 0.125125125..., which can be represented exactly as a rational number rather than approximated. Given Data / Assumptions: The decimal given is 0.125125125..., where the block "125" repeats indefinitely. We must express this as a fraction in simplest form. No rounding is involved, since the repeating pattern is exact. Concept / Approach: For a decimal where a block of three digits repeats, we multiply the number by 1000 to shift the decimal three places to the right. Subtracting the original number from this new value cancels the repeating part. This yields a simple linear equation whose solution is the required fraction. Finally, we check whether the fraction can be simplified further. Step-by-Step Solution: Step 1: Let x = 0.125125125... Step 2: Multiply both sides by 1000: 1000x = 125.125125... Step 3: Subtract the original x: 1000x - x = 125.125125... - 0.125125... Step 4: The recurring parts cancel, giving 999x = 125. Step 5: Solve for x: x = 125 / 999. Step 6: Check if 125 / 999 can be simplified. The prime factors of 125 are 5 × 5 × 5, and 999 = 3 × 3 × 3 × 37. There is no common factor, so the fraction is already in simplest form. Verification / Alternative check: Divide 125 by 999 using long division. You will obtain 0.125125125..., where "125" repeats continuously, confirming that 125 / 999 is the correct rational representation of the decimal. Why Other Options Are Wrong: 125/99 corresponds to a decimal with a shorter repeating block and a different value. 125/9999 would represent a much smaller decimal since the denominator is larger. 125/9 is approximately 13.888..., clearly not between 0 and 1. 25/198 simplifies to 25 / (2 × 99) = about 0.12626..., which is not equal to the repeating 0.125125125.... Common Pitfalls: One common mistake is choosing 125/99 by confusing the number of digits in the repeating block. Since "125" has three digits, you must multiply by 1000, not 100, leading to the denominator 999 instead of 99. Forgetting to simplify the fraction when possible is another error, though in this case 125 / 999 is already in lowest terms. Final Answer: The recurring decimal 0.125125125... is exactly equal to the fraction 125/999.

Question 7

The product 54.327 × 357.2 × 0.0057 is to be rewritten by shifting decimal points only. Which of the following rewritten products is exactly equal to this value?

Accepted Answer

Introduction / Context: This question focuses on handling decimal numbers by shifting decimal points while keeping the overall product unchanged. It tests your understanding that multiplying or dividing by 10, 100, and similar powers of 10 can be balanced across factors so that the final result remains the same. Given Data / Assumptions: Original product: 54.327 × 357.2 × 0.0057. We must choose an equivalent product formed by shifting decimal points. No rounding or change in the actual numerical value is allowed. Concept / Approach: Shifting a decimal point to the left divides a number by a power of 10, and shifting it to the right multiplies it by a power of 10. Any decrease in one factor can be compensated by a corresponding increase in another factor so that the total product remains constant. We can compare numerically or, more simply, express the decimals using powers of 10. Step-by-Step Solution: Step 1: Compute or inspect the original product numerically for reference. Step 2: Consider option A: 5.4327 × 3.572 × 5.7. Step 3: Note that 54.327 = 5.4327 × 10 and 357.2 = 3.572 × 100, while 0.0057 = 5.7 / 1000. Step 4: The combined power of 10 in the original product is 10 × 100 × (1 / 1000) = 1, so there is no net effect from the powers of 10. Step 5: Therefore, 54.327 × 357.2 × 0.0057 = (5.4327 × 10) × (3.572 × 100) × (5.7 / 1000). Step 6: Rearranging, this equals 5.4327 × 3.572 × 5.7 × (10 × 100 / 1000) = 5.4327 × 3.572 × 5.7. Step 7: Thus, option A is exactly equal to the original product. Verification / Alternative check: You may calculate the numerical value of 54.327 × 357.2 × 0.0057 and 5.4327 × 3.572 × 5.7 using a calculator. Both will yield the same decimal value (approximately 110.61194508), confirming the equality. Why Other Options Are Wrong: Option B uses 0.57 instead of 5.7, which introduces an extra division by 10, making the product ten times smaller. Option C uses 0.0000057 along with large integers, resulting in an overall product that is ten times larger than the original. Option E shifts decimal points in a way that changes the net power of 10 and therefore does not preserve the product. Option D "None of these" is incorrect because option A is a valid equivalent expression. Common Pitfalls: A common mistake is to shift decimal points in several numbers without checking the combined effect of powers of 10. It is important to track how many times you effectively multiply or divide by 10 overall. Another pitfall is relying only on approximate mental calculations instead of checking powers of 10 systematically. Final Answer: The expression that is exactly equal to the original product is 5.4327 x 3.572 x 5.7.

Question 8

Karthik read 6/13 of a book in the first week and 5/9 of the remaining part of the book in the second week. If about 100 pages were still unread after the second week, approximately how many pages were there in the entire book?

Accepted Answer

Introduction / Context: This question tests proportional reasoning with fractions applied in a real-life context of reading a book. The reader finishes fractional parts in two weeks, and a small known number of pages remains unread. You need to model these fractional readings algebraically and then estimate the total number of pages in the book. Given Data / Assumptions: Let the total number of pages in the book be N. Karthik reads 6/13 of the book in the first week. In the second week, he reads 5/9 of the remaining pages. After the second week, around 100 pages are still unread. The question asks for an approximate total number of pages. Concept / Approach: We successively remove fractions of the total: first 6/13 of N, then 5/9 of what remains. The unread portion is the fraction that remains after both weeks. We set this remaining amount approximately equal to 100 and solve for N. Finally, we choose the option that gives a leftover closest to 100 pages. Step-by-Step Solution: Step 1: After week one, pages read = (6/13)N, so pages remaining = N - (6/13)N = (7/13)N. Step 2: In week two, Karthik reads 5/9 of the remaining pages, so pages read in week two = (5/9) × (7/13)N. Step 3: Remaining after week two = (4/9) × (7/13)N = (28/117)N. Step 4: Set this remaining amount close to 100 pages: (28/117)N ≈ 100. Step 5: Solve for N: N ≈ 100 × (117 / 28) ≈ 417.86 pages. Step 6: Among the options 404, 415, 418, 420, 432, the value 418 is closest to 417.86. Verification / Alternative check: For N = 418 pages, remaining pages after both weeks are: (28/117) × 418 ≈ 100.03 pages, which is essentially 100 pages when rounded to the nearest whole number. This confirms that 418 is the best approximate answer. Why Other Options Are Wrong: 404: Gives remaining pages significantly below 100 (around 96.7 pages). 415: Leaves roughly 99.3 pages, slightly less accurate than 418 in matching 100 pages. 420: Leaves about 100.5 pages, a bit further away from 100 than 418 is. 432: Leaves more than 100 pages, not the closest to the target 100. Common Pitfalls: Many learners mistakenly apply the fraction 5/9 to the original total instead of the remaining portion after the first week. Others forget that the problem explicitly asks for an approximate value, so they may be uncomfortable with a non-integer intermediate result. Recognising that real-life quantities like book pages in word problems can sometimes be approximated is important in aptitude tests. Final Answer: The total number of pages in the book is approximately 418 pages.

Question 9

If two proper fractions (each greater than 0 and less than 1) are multiplied together, how does the value of their product compare with each of the original fractions?

Accepted Answer

Introduction / Context: This is a conceptual question about the behaviour of fractions when multiplied. Specifically, it focuses on proper fractions, which are fractions between 0 and 1. Understanding how products of such numbers behave is fundamental in arithmetic and number theory, and it often appears in competitive exams to test conceptual clarity. Given Data / Assumptions: We are dealing with two proper fractions, say a and b, where 0 < a < 1 and 0 < b < 1. We consider the product a × b. We must compare a × b with each of the original fractions a and b. Concept / Approach: For any number between 0 and 1, multiplying it by a positive number less than 1 makes it smaller. This is because you are taking a fraction of it. Therefore, when you multiply a by b (where b is less than 1), the product is less than a. Similarly, multiplying b by a (where a is less than 1) makes the product less than b. Hence, the product is less than both a and b. Step-by-Step Solution: Step 1: Assume 0 < a < 1 and 0 < b < 1. Step 2: Because b is less than 1 and positive, a × b < a. Step 3: Similarly, because a is less than 1 and positive, a × b < b. Step 4: Therefore, a × b is less than both a and b. Step 5: This is true for all positive proper fractions a and b. Verification / Alternative check: Take example fractions: a = 1/2 and b = 2/3. Then a × b = (1/2) × (2/3) = 1/3. Notice that 1/3 < 1/2 and 1/3 < 2/3, confirming the general rule. Try other pairs, such as 3/4 and 4/5, and you will observe the same pattern. Why Other Options Are Wrong: Always greater than either: This would require the product of numbers less than 1 to be larger, which contradicts basic arithmetic. Sometimes greater and sometimes less: For positive proper fractions, the product is always less than each factor, so there is no variation in behaviour. Remains the same: The product equals a only if b = 1, and equals b only if a = 1, but proper fractions are strictly less than 1, so this cannot happen here. Cannot be determined: In fact, it can be determined using general reasoning as shown. Common Pitfalls: Many learners confuse this with multiplication involving numbers greater than 1 or mixing fractions greater and less than 1. It is essential to focus on the condition that both fractions are strictly between 0 and 1. Visualising the fractions on a number line can help build intuition that their product must lie closer to 0 than either factor. Final Answer: The product of two proper fractions is always less than either of the original fractions.

Question 10

When the fraction 18/1000 is written as a decimal number in simplest form, what is the resulting quotient?

Accepted Answer

Introduction / Context: Converting fractions into decimal form is a basic but essential skill in arithmetic and aptitude tests. In this question, you must convert 18/1000 into a decimal and recognise its simplest decimal representation. Accuracy in placing the decimal point is crucial here. Given Data / Assumptions: Fraction given: 18/1000. We must express this as a decimal number. The decimal must be in simplest form (no unnecessary trailing zeros). Concept / Approach: A fraction with denominator 1000 can be written as a decimal by moving the decimal point three places to the left in the numerator. This is because 1000 = 10^3. After writing the decimal, we can remove trailing zeros to obtain the simplest form. Step-by-Step Solution: Step 1: Start with the fraction 18/1000. Step 2: Recognise that dividing by 1000 moves the decimal three places to the left. Step 3: Write 18 as 18.000 and move the decimal three places: 18 ÷ 1000 = 0.018. Step 4: The decimal 0.018 has no unnecessary trailing zeros, so it is already in simplest form. Verification / Alternative check: We can simplify the fraction first: 18/1000. Divide numerator and denominator by 2 to get 9/500. Now compute 9/500 as a decimal: 9 ÷ 500 = 9 ÷ (5 × 100) = (9/5) ÷ 100 = 1.8 ÷ 100 = 0.018. This confirms the same decimal representation. Why Other Options Are Wrong: 1.8 corresponds to 18/10, not 18/1000. 0.18 corresponds to 18/100, which is ten times larger than 18/1000. 0.0018 corresponds to 1.8/1000 or 18/10000, which is ten times smaller than 18/1000. 0.00018 is even smaller, equal to 18/100000, and does not match the given fraction. Common Pitfalls: The main error occurs in misplacing the decimal point, often moving it only one or two places for a denominator of 1000 instead of three. Another mistake is to forget to reduce the fraction before converting, although in this case both methods lead to the same decimal. Careful attention to the number of zeros in the denominator helps avoid such mistakes. Final Answer: The decimal representation of 18/1000 is 0.018.

Question 11

For which value of x will the expression 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02 become a perfect square of a binomial?

Accepted Answer

Introduction / Context: This problem examines your ability to recognise patterns in algebraic expressions, especially perfect square trinomials. A perfect square trinomial can be written as (a + b)^2 or (a - b)^2. Here, decimals are involved, but the same pattern recognition applies directly. Given Data / Assumptions: Expression: 11.98 × 11.98 + 11.98 × x + 0.02 × 0.02. We must find x such that the entire expression is a perfect square of a binomial. All multiplications are ordinary real-number multiplications. Concept / Approach: Recall the identity (a + b)^2 = a^2 + 2ab + b^2. To match our expression with this form, we can let a = 11.98 and b = 0.02. Then a^2 corresponds to 11.98 × 11.98 and b^2 corresponds exactly to 0.02 × 0.02. The middle term 11.98 × x must match 2ab = 2 × 11.98 × 0.02. This condition allows us to solve for x. Step-by-Step Solution: Step 1: Set a = 11.98 and b = 0.02. Step 2: Identify a^2 = 11.98 × 11.98, which is present in the expression. Step 3: Identify b^2 = 0.02 × 0.02, which is also present. Step 4: For a perfect square, the middle term must be 2ab = 2 × 11.98 × 0.02. Step 5: Compute 2ab: 2 × 11.98 × 0.02 = 11.98 × 0.04. Step 6: The given middle term is 11.98 × x, so we require 11.98 × x = 11.98 × 0.04. Step 7: Cancel 11.98 (nonzero) from both sides to get x = 0.04. Verification / Alternative check: Substitute x = 0.04 into the expression: 11.98 × 11.98 + 11.98 × 0.04 + 0.02 × 0.02 = a^2 + 2ab + b^2, where a = 11.98 and b = 0.02. This equals (11.98 + 0.02)^2 = 12.00^2 = 144, clearly a perfect square. So x = 0.04 works perfectly. Why Other Options Are Wrong: 0.01, 0.02, and 0.03 all give middle terms 11.98 × x that are smaller than the required 2ab. They do not satisfy the condition 11.98 × x = 11.98 × 0.04. 0.4 makes the middle term too large and completely breaks the perfect square pattern. Common Pitfalls: Learners sometimes confuse 2ab with a × b or attempt to match the expression with (a - b)^2 without checking signs. Another common mistake is to be intimidated by decimals. Treating 11.98 and 0.02 symbolically as a and b simplifies the reasoning, and decimals can be handled at the end if necessary. Final Answer: The expression becomes a perfect square when x = 0.04.

Question 12

A person was asked to state his age in years. He replied: Take my age 3 years from now, multiply it by 3, then subtract 3 times my age 3 years ago, and the result will equal my present age. How old is this person now?

Accepted Answer

Introduction / Context: This is a classic age puzzle that tests your ability to translate a verbal description into an algebraic equation. Once the equation is formed, solving it is straightforward. Such puzzles frequently appear in aptitude exams to evaluate logical thinking and algebra skills. Given Data / Assumptions: Let the present age of the person be A years. Age 3 years from now is A + 3. Age 3 years ago was A - 3. The statement says: 3 × (age 3 years from now) minus 3 × (age 3 years ago) equals the present age A. Concept / Approach: We convert the language into an algebraic equation. Then we simplify the equation and solve for A. As the numbers are small and linear, it is a simple equation in one variable. Step-by-Step Solution: Step 1: Assume present age = A years. Step 2: Age 3 years from now = A + 3. Step 3: Age 3 years ago = A - 3. Step 4: According to the condition: 3 × (A + 3) - 3 × (A - 3) = A. Step 5: Expand the left side: 3A + 9 - (3A - 9) = A. Step 6: Simplify: 3A + 9 - 3A + 9 = A, so 18 = A. Step 7: Therefore, the present age of the person is 18 years. Verification / Alternative check: Check the condition with A = 18. Age 3 years from now: 21. Age 3 years ago: 15. 3 × 21 = 63 and 3 × 15 = 45. The difference is 63 - 45 = 18, which equals the present age. This confirms the solution. Why Other Options Are Wrong: 20: If A = 20, the calculation would give 3 × 23 - 3 × 17 = 69 - 51 = 18, not 20. 24: If A = 24, 3 × 27 - 3 × 21 = 81 - 63 = 18, not 24. 30: If A = 30, 3 × 33 - 3 × 27 = 99 - 81 = 18, not 30. 22: Similarly gives a result different from 22. Only 18 satisfies the equation. Common Pitfalls: The main difficulty is correctly interpreting the verbal description. Some learners may forget the negative sign when subtracting 3 × (A - 3) or may misread the phrase "3 years ago." Writing each step of the equation explicitly and expanding carefully prevents sign errors. Final Answer: The person is currently 18 years old.

Question 13

Which of the following sequences of fractions is correctly arranged in descending order (from greatest value to least value)?

Accepted Answer

Introduction / Context: Ordering fractions is a common task in aptitude and school mathematics. This question asks you to determine which sequence lists the given fractions in descending order, that is, from the largest value to the smallest. Understanding how to compare fractions is essential to solve such problems accurately. Given Data / Assumptions: Fractions involved: 5/9, 7/11, 8/15, 11/17. We must identify the option that lists these fractions from greatest to least. No calculators are strictly required; approximate values or cross-multiplication are enough. Concept / Approach: There are two main ways to compare these fractions: Convert each fraction to a decimal approximation. Use cross-multiplication to compare pairs of fractions without explicitly using decimals. We will use decimal approximations for clarity and then confirm the order. Step-by-Step Solution: Step 1: Approximate each fraction as a decimal: 5/9 ≈ 0.5556 7/11 ≈ 0.6364 8/15 ≈ 0.5333 11/17 ≈ 0.6471 Step 2: Arrange these decimal values from largest to smallest: 0.6471 (11/17), 0.6364 (7/11), 0.5556 (5/9), 0.5333 (8/15). Step 3: Convert this back into a sequence of fractions: 11/17, then 7/11, then 5/9, and finally 8/15. Verification / Alternative check: We can compare 11/17 and 7/11 by cross-multiplication: 11 × 11 = 121 and 7 × 17 = 119. Since 121 > 119, 11/17 > 7/11. Similarly, compare 7/11 and 5/9: 7 × 9 = 63 and 5 × 11 = 55, so 7/11 > 5/9. Finally, 5/9 and 8/15: 5 × 15 = 75 and 8 × 9 = 72, so 5/9 > 8/15. This confirms the descending order 11/17 > 7/11 > 5/9 > 8/15. Why Other Options Are Wrong: Options A, B, and C place at least one fraction out of order when compared carefully using decimals or cross-multiplication. Option E "None of these" is incorrect because option D provides the correct descending order. Common Pitfalls: A frequent mistake is to compare fractions by looking only at numerators or denominators separately, which can be misleading. Another issue is rounding decimals too aggressively, which might reverse the order of close values. Using cross-multiplication is a safe way to avoid these errors. Final Answer: The correct descending order is 11/17, 7/11, 5/9, 8/15.

Question 14

When 52.416 is divided by 0.0168, what is the resulting quotient, given that 52,416 ÷ 312 = 168?

Accepted Answer

Introduction / Context: This problem combines decimal division with a useful shortcut based on scaling by powers of 10. You are given one division fact (52416 ÷ 312 = 168) and asked to compute a related quotient involving decimals. Recognising how to relate the two problems is the key skill being tested. Given Data / Assumptions: Given: 52,416 ÷ 312 = 168. Required: 52.416 ÷ 0.0168. All operations are standard real-number divisions. Concept / Approach: We can eliminate decimals by multiplying both the dividend and the divisor by the same power of 10. This does not change the value of the quotient. Once the decimals are removed, we may be able to relate the resulting division to a simple integer division that is easy to compute. Step-by-Step Solution: Step 1: Consider the target expression: 52.416 ÷ 0.0168. Step 2: Multiply both dividend and divisor by 1000 to remove three decimal places: (52.416 × 1000) ÷ (0.0168 × 1000). Step 3: This becomes 52416 ÷ 16.8. Step 4: Multiply both numbers by 10 to remove the remaining decimal: (52416 × 10) ÷ (16.8 × 10) = 524160 ÷ 168. Step 5: Now evaluate 524160 ÷ 168. Notice that 524160 = 3120 × 168. Step 6: Therefore, 524160 ÷ 168 = 3120. Step 7: Hence, 52.416 ÷ 0.0168 = 3120. Verification / Alternative check: Use the given relationship. Since 52416 ÷ 312 = 168, we can try to express 52.416 ÷ 0.0168 in a way related to 168. Recognise that dividing by 0.0168 is the same as multiplying by its reciprocal. Adjusting by factors of 10 eventually leads to the same result 3120. A calculator check will also confirm that 52.416 ÷ 0.0168 = 3120 exactly. Why Other Options Are Wrong: 3.12 and 312 are much smaller than the true quotient, resulting from moving the decimal point incorrectly or not accounting for all the powers of 10. 31200 is ten times larger than the correct result, likely from overcompensating when clearing decimals. None of these is incorrect because 3120 is a valid and exact answer. Common Pitfalls: The most common errors arise from mishandling decimal places. If you do not multiply both numerator and denominator by the same power of 10, you accidentally change the quotient. Another pitfall is stopping halfway, for example after removing only some of the decimal places, which makes the division more complicated than necessary. Final Answer: The quotient of 52.416 divided by 0.0168 is 3120.

Question 15

Evaluate the expression (0.2 × 0.2 + 0.01) × (0.1 × 0.1 + 0.02)^(-1) and express your answer as a simplified fraction.

Accepted Answer

Introduction / Context: This question assesses your ability to work with decimals, basic algebraic expressions, and reciprocals. The notation ( )^(-1) means the reciprocal of the quantity inside the brackets. After evaluating the decimal operations, you must simplify the result to a fraction in lowest terms. Given Data / Assumptions: Expression: (0.2 × 0.2 + 0.01) × (0.1 × 0.1 + 0.02)^(-1). We must compute this exactly. The final answer should be given as a simplified fraction. Concept / Approach: First, simplify each part inside the brackets. Then evaluate the reciprocal of the second bracket. Finally, multiply the simplified first bracket by this reciprocal. Converting the decimals to fractions at some stage makes simplification easier and more precise. Step-by-Step Solution: Step 1: Simplify the first bracket: 0.2 × 0.2 + 0.01. Step 2: Compute 0.2 × 0.2 = 0.04. Step 3: Add 0.01 to get 0.04 + 0.01 = 0.05. Step 4: Simplify the second bracket: 0.1 × 0.1 + 0.02. Step 5: Compute 0.1 × 0.1 = 0.01. Step 6: Add 0.02 to get 0.01 + 0.02 = 0.03. Step 7: Apply the power (−1): (0.03)^(-1) = 1 / 0.03. Step 8: Now the whole expression is 0.05 × (1 / 0.03) = 0.05 / 0.03. Step 9: Convert to fractions: 0.05 = 5/100 and 0.03 = 3/100. Step 10: Compute (5/100) / (3/100) = (5/100) × (100/3) = 5/3. Verification / Alternative check: You may compute 0.05 ÷ 0.03 directly as a decimal: 0.05 ÷ 0.03 ≈ 1.6666..., which corresponds to 5/3. This matches the fraction obtained via symbolic manipulation, confirming the answer. Why Other Options Are Wrong: 9/3 = 3, which is too large and would come from miscalculating 0.05 or 0.03. 11/3 ≈ 3.6667 and 13/3 ≈ 4.3333, both far from the correct value around 1.67. 1 would require 0.05 and 0.03 to be equal, which is not the case. Common Pitfalls: A common mistake is to ignore the meaning of the power (−1) and treat it like a negative exponent in another context instead of a reciprocal. Another pitfall is rough decimal rounding, such as treating 0.05 / 0.03 as 5 / 3 without justifying the intermediate steps. Converting decimals to fractions is a clean way to ensure accuracy. Final Answer: The value of the expression is 5/3.

Question 16

What is the decimal equivalent (decimal conversion) of the fraction 1/8 ?

Accepted Answer

Introduction / Context: Converting simple fractions into decimals is a fundamental skill in arithmetic and is frequently tested in aptitude examinations. The fraction 1/8 is a standard benchmark value, and knowing its decimal form helps in many calculations involving percentages, ratios, and probability. Given Data / Assumptions: We are given the fraction 1/8. We must convert it into a decimal number. The answer should be exact, with no rounding. Concept / Approach: To convert a fraction to a decimal, we divide the numerator by the denominator. For 1/8, this means performing 1 ÷ 8. Since 8 does not divide 1 directly, we introduce decimal places and continue the division until it terminates or repeats. In this case, the decimal terminates after three places. Step-by-Step Solution: Step 1: Write the division 1 ÷ 8. Step 2: Since 1 is smaller than 8, write 1.000 and think of dividing 10, 100, and 1000 by 8 step by step. Step 3: 8 goes into 10 once, giving 1 with a remainder of 2 (since 10 − 8 = 2). So the first decimal digit is 1. Step 4: Bring down another 0 to make 20. 8 goes into 20 two times (2 × 8 = 16), remainder 4. The second decimal digit is 2. Step 5: Bring down another 0 to make 40. 8 goes into 40 exactly 5 times, remainder 0. The third decimal digit is 5. Step 6: The division ends with no remainder, so 1 ÷ 8 = 0.125. Verification / Alternative check: Multiply 0.125 by 8 to ensure it returns 1. Compute 0.125 × 8 = 1.000, confirming that 0.125 is exactly equal to 1/8. This verifies the conversion. Why Other Options Are Wrong: 0.375 is the decimal for 3/8, not 1/8. 0.247 and 0.214 do not correspond to any simple fraction like 1/8 and appear as distractor values. 0.875 corresponds to 7/8, not 1/8. Common Pitfalls: Some learners confuse 1/8 with similar-looking fractions like 1/4 or 3/8 and misremember the decimal equivalents. Others may stop division too early and round incorrectly. Practising a few standard fraction-decimal pairs, such as 1/2, 1/4, 1/5, 1/8, and 3/8, helps build strong numerical intuition. Final Answer: The decimal equivalent of 1/8 is 0.125.

Question 17

In basic fraction simplification, what is the fraction 125/1000 written in its simplest form?

Accepted Answer

Introduction:This question checks your understanding of how to reduce a fraction to its simplest or lowest terms. Simplifying fractions is a fundamental skill in arithmetic and is used frequently in higher mathematics, data interpretation, and everyday calculations. Given Data / Assumptions: Given fraction: 125/1000 We must express this fraction in its simplest form. Simplest form means numerator and denominator have no common factor greater than 1. Concept / Approach:To simplify a fraction, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest positive integer that divides both numbers exactly. After dividing by the GCD, the resulting fraction is in lowest terms. The value of the fraction does not change, only its representation becomes simpler. Step-by-Step Solution:Step 1: Identify the fraction: 125/1000.Step 2: Find the greatest common divisor of 125 and 1000.Step 3: Recognize that 125 is a factor of 1000, because 125 * 8 = 1000.Step 4: Therefore, GCD(125, 1000) = 125.Step 5: Divide numerator and denominator by 125.Step 6: Numerator: 125 / 125 = 1.Step 7: Denominator: 1000 / 125 = 8.Step 8: The simplified fraction is 1/8. Verification / Alternative check:You can convert both 125/1000 and 1/8 to decimals. 125/1000 = 0.125. Also, 1/8 = 0.125. Since the decimal values are the same, the fractions are equivalent, confirming that 1/8 is the correct simplified form of 125/1000. Why Other Options Are Wrong:2/9, 3/8, and 4/7 all represent different decimal values (approximately 0.222..., 0.375, and 0.5714 respectively), none of which equal 0.125. They are included as plausible looking fractions but do not match the given value when simplified or converted to decimals. Common Pitfalls:One common mistake is to divide only by a small common factor instead of the greatest one, for example dividing 125/1000 by 5 or 25 and stopping too early. While repeated simplification works, using the greatest common divisor is faster and ensures the fraction is fully simplified in one step. Final Answer:The fraction 125/1000 in simplest form is 1/8.

Question 18

When the fraction 5/8 is converted to a decimal number, what is its exact decimal value?

Accepted Answer

Introduction:This problem tests your ability to convert a simple fraction into its decimal form. Understanding how to move between fractions and decimals is essential in arithmetic, percentages, and many aptitude questions. Given Data / Assumptions: Given fraction: 5/8 We must express 5/8 as a decimal. Assume exact arithmetic with no rounding. Concept / Approach:A fraction a/b can be converted to a decimal by performing the division a ÷ b. If the denominator has only prime factors 2 and 5, the decimal will terminate. Since 8 = 2 * 2 * 2, we expect a terminating decimal. We simply divide 5 by 8 using normal long division or mental math. Step-by-Step Solution:Step 1: Write the fraction as a division: 5/8 = 5 ÷ 8.Step 2: Since 5 is smaller than 8, think of 5.000 ÷ 8.Step 3: 8 goes into 50 six times because 6 * 8 = 48.Step 4: Subtract 48 from 50 to get a remainder of 2, then bring down the next 0 to make 20.Step 5: 8 goes into 20 two times because 2 * 8 = 16. Remainder is 4, bring down another 0 to make 40.Step 6: 8 goes into 40 five times because 5 * 8 = 40. Remainder is 0, so the division ends.Step 7: Reading the digits, we get 0.625. Verification / Alternative check:Multiply the decimal back by 8 to check: 0.625 * 8 = 5. This confirms that 0.625 is exactly equal to 5/8, so the conversion is correct. Why Other Options Are Wrong:0.541, 0.258, 0.147, and 0.600 yield values different from 5/8 when multiplied by 8. For example, 0.600 * 8 = 4.8, which is less than 5, so it cannot represent 5/8. The other decimals are even further away from 0.625. Common Pitfalls:Learners sometimes try to guess decimals instead of performing the actual division, which leads to small but important errors. Another mistake is stopping the division too early or misplacing the decimal point. Carefully performing stepwise division avoids these issues. Final Answer:The decimal equivalent of 5/8 is 0.625.

Question 19

Evaluate the decimal expression (0.2 * 0.2 + 0.01) * (0.1 * 0.1 + 0.02)^(-1). What is the resulting value expressed as a simplified fraction?

Accepted Answer

Introduction:This question checks your ability to handle decimal arithmetic, bracket operations, and negative powers. The expression involves multiplication, addition, and an inverse (power of minus one), and finally asks for the answer in fractional form. Given Data / Assumptions: Expression: (0.2 * 0.2 + 0.01) * (0.1 * 0.1 + 0.02)^(-1) All numbers are ordinary decimals. Power -1 means reciprocal (or inverse) of the quantity inside the brackets. Concept / Approach:We simplify inside each bracket first. Then we apply the negative power by taking the reciprocal of that result. Finally, we multiply the two parts. Working carefully with decimal multiplication and addition is crucial. Converting intermediate decimals to fractions can make simplification easier and help in obtaining the final answer as a reduced fraction. Step-by-Step Solution:Step 1: Compute 0.2 * 0.2 = 0.04.Step 2: Add 0.01: 0.04 + 0.01 = 0.05.Step 3: Compute 0.1 * 0.1 = 0.01.Step 4: Add 0.02: 0.01 + 0.02 = 0.03.Step 5: Raise the second bracket to the power -1: (0.03)^(-1) = 1 / 0.03.Step 6: Now the whole expression becomes 0.05 * (1 / 0.03) = 0.05 / 0.03.Step 7: Convert decimals to fractions: 0.05 = 5/100 and 0.03 = 3/100.Step 8: Compute (5/100) / (3/100) = (5/100) * (100/3) = 5/3.So the simplified result is 5/3. Verification / Alternative check:You can use decimal arithmetic: 0.05 ÷ 0.03 ≈ 1.6666..., and 5/3 as a decimal is 1.6666..., which matches. This confirms the correctness of the fractional result. Why Other Options Are Wrong:9/3 equals 3, 11/3 is approximately 3.67, 13/3 is about 4.33, and 7/3 is approximately 2.33. None of these match the decimal 1.6666..., so they cannot be correct values for the given expression. Common Pitfalls:Typical mistakes include misinterpreting the power -1 as multiplication by -1 instead of taking the reciprocal, or adding and multiplying decimals incorrectly. Another frequent error is to divide 3 by 5 instead of 5 by 3 when simplifying 0.05/0.03. Converting decimals to fractions and handling the arithmetic step by step helps avoid confusion. Final Answer:The value of the expression is 5/3.

Question 20

When the decimal 0.75 is written as a vulgar fraction in its simplest form, what is that fraction?

Accepted Answer

Introduction:This problem asks you to convert a terminating decimal into a vulgar fraction (ordinary fraction) in simplest form. Converting between decimals and fractions is an essential numerical skill used in percentages, ratios, and many quantitative aptitude questions. Given Data / Assumptions: Decimal given: 0.75 We need the equivalent fraction. The answer should be in lowest terms. Concept / Approach:A decimal like 0.75 can be written as a fraction by reading it as seventy five hundredths, which is 75/100. After that, we reduce the fraction by dividing numerator and denominator by their greatest common divisor. Since 0.75 is a terminating decimal, this process is straightforward. Step-by-Step Solution:Step 1: Write 0.75 as a fraction: 0.75 = 75/100.Step 2: Find the greatest common divisor of 75 and 100.Step 3: Both numbers are divisible by 25: 75 = 25 * 3 and 100 = 25 * 4.Step 4: Divide numerator and denominator by 25.Step 5: Numerator: 75 / 25 = 3.Step 6: Denominator: 100 / 25 = 4.Step 7: Therefore, 0.75 = 3/4 in simplest form. Verification / Alternative check:Convert 3/4 back to decimal: 3 ÷ 4 = 0.75. This confirms that 3/4 is indeed the correct fraction for 0.75. No further simplification is possible because 3 and 4 have no common factor greater than 1. Why Other Options Are Wrong:3/5 is 0.6, which is less than 0.75. 3/2 equals 1.5, which is greater than 0.75. 3/7 is about 0.4286, and 1/4 is 0.25; both differ significantly from 0.75. Only 3/4 matches the decimal 0.75 exactly. Common Pitfalls:Students sometimes misread 0.75 as 75/1000 or confuse it with 0.7 or 0.8. Another frequent error is to stop simplifying at 75/100 and forget to reduce it. Always remember to divide by the greatest common divisor to obtain a fraction in its lowest terms. Final Answer:The decimal 0.75 written as a vulgar fraction in simplest form is 3/4.

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