Question 1

Among the fractions 2/3, 3/4, 4/5 and 5/6, what is the difference between the largest fraction and the smallest fraction?

Accepted Answer

Introduction:This question tests your understanding of comparing fractions and finding the numerical difference between them. Being able to order fractions and compute differences is important for solving ratio, proportion, and percentage problems in aptitude tests. Given Data / Assumptions: Fractions given: 2/3, 3/4, 4/5, 5/6. We must identify the largest and smallest fraction. Then, compute: largest fraction minus smallest fraction. Concept / Approach:There are two practical methods: convert each fraction to a decimal or bring them to a common denominator. For quick comparisons, approximate decimals are often easiest. Once we know which is smallest and which is largest, we subtract them using fraction arithmetic to obtain the exact difference. Step-by-Step Solution:Step 1: Convert each fraction to decimal for comparison.2/3 ≈ 0.6667.3/4 = 0.75.4/5 = 0.8.5/6 ≈ 0.8333.Step 2: Identify smallest and largest.The smallest value is 2/3 (≈ 0.6667).The largest value is 5/6 (≈ 0.8333).Step 3: Compute the difference: 5/6 − 2/3.Step 4: Express 2/3 with denominator 6: 2/3 = 4/6.Step 5: Now subtract: 5/6 − 4/6 = (5 − 4)/6 = 1/6.So the difference is 1/6. Verification / Alternative check:Using decimals, approximate difference is 0.8333 − 0.6667 ≈ 0.1666. The fraction 1/6 is equal to approximately 0.1666, which confirms that the exact fractional difference is 1/6. Why Other Options Are Wrong:1/12 is about 0.0833, which is only half the actual difference. 1/20 is 0.05 and 1/30 is about 0.0333, both too small. 1/15 is about 0.0667, also too small. Only 1/6 matches the calculated difference between 5/6 and 2/3. Common Pitfalls:Some learners may incorrectly assume that 2/3 is larger than 5/6 or compare only numerators without considering denominators. Others may make errors in converting to decimals or subtracting fractions with unlike denominators. Always use a consistent method, such as converting to a common denominator or accurately using decimal approximations. Final Answer:The difference between the largest and smallest fractions is 1/6.

Question 2

Evaluate 138.009 + 341.981 - 146.305 and find the value of the missing term in the equation 138.009 + 341.981 - 146.305 = 123.6 + x. What is the value of x?

Accepted Answer

Introduction:This question checks your comfort with decimal arithmetic and simple equation balancing. You are given a numerical expression and an equation involving a missing value x. The goal is to evaluate the left side and then solve for x on the right side. Given Data / Assumptions: Expression: 138.009 + 341.981 − 146.305. Equation: 138.009 + 341.981 − 146.305 = 123.6 + x. We must find the value of x that satisfies this equation. Concept / Approach:First, compute the value of the left hand side by adding and subtracting the decimals. Then equate this result to 123.6 + x. Solving for x is done by subtracting 123.6 from the total. Working carefully with decimal places ensures accuracy. Step-by-Step Solution:Step 1: Add the first two numbers: 138.009 + 341.981.Step 2: 138.009 + 341.981 = 479.990 (since 0.009 + 0.981 = 0.990 and 138 + 341 = 479).Step 3: Subtract 146.305 from this sum: 479.990 − 146.305.Step 4: Perform the subtraction: 479.990 − 146.305 = 333.685.Step 5: Now the equation becomes 333.685 = 123.6 + x.Step 6: To find x, subtract 123.6 from 333.685.Step 7: 333.685 − 123.600 = 210.085.Step 8: Therefore x = 210.085. Verification / Alternative check:Substitute x = 210.085 back into the right side: 123.6 + 210.085 = 333.685, which matches the left side value. This confirms that the solution is correct and consistent. Why Other Options Are Wrong:199.57, 295.05, and 120.085 do not satisfy the equation when substituted for x, because 123.6 plus any of these values does not equal 333.685. The option “None of these” is incorrect because a valid correct value, 210.085, is present among the options. Common Pitfalls:Errors often arise from misalignment of decimal points during addition and subtraction. Another common mistake is forgetting to subtract 123.6 from the total to isolate x. Writing the intermediate steps clearly and keeping decimal places lined up helps avoid such errors. Final Answer:The value of x that makes the equation true is 210.085.

Question 3

What is the decimal representation of the fraction 1/5 (that is, one divided by five)?

Accepted Answer

Introduction:This question asks you to convert a simple fraction, 1/5, into its decimal form. Such conversions are widely used in percentage problems, ratios, and everyday calculations where decimals are more convenient than fractions. Given Data / Assumptions: Fraction: 1/5. We must express 1/5 as a decimal. We seek an exact, not approximate, decimal value. Concept / Approach:To convert a fraction a/b to a decimal, we perform the division a ÷ b. Because 5 is a factor of 10, 1/5 turns into a terminating decimal. Alternatively, we can use the known fact that 0.2 is equal to 1/5, but it is useful to see the division process clearly. Step-by-Step Solution:Step 1: Interpret the fraction as division: 1/5 = 1 ÷ 5.Step 2: Since 1 is smaller than 5, write it as 1.0 and then 1.00 as needed.Step 3: 5 goes into 10 exactly 2 times, because 2 * 5 = 10.Step 4: So 1.0 ÷ 5 = 0.2 exactly.Step 5: Writing this with three decimal places as in one of the options, 0.2 is the same as 0.200.Thus, the decimal representation is 0.200. Verification / Alternative check:Check by converting 0.2 back to a fraction: 0.2 = 2/10, and 2/10 simplifies to 1/5 when both numerator and denominator are divided by 2. This confirms that 0.2 (or 0.200) is exactly 1/5. Why Other Options Are Wrong:0.333 corresponds to approximately 1/3, not 1/5. 0.666 is about 2/3. 0.452 does not correspond to a simple fraction like 1/5. 0.500 is equal to 1/2, which is larger than 1/5. Only 0.200 matches the correct decimal for 1/5. Common Pitfalls:Some learners confuse 1/5 with 1/4 or 1/3 and may guess 0.25 or 0.333. Others may incorrectly think 1/5 equals 0.5 because of the number 5. Always remember that 1/5 means one part out of five equal parts, which is smaller than half, so its decimal must be less than 0.5. Final Answer:The fraction 1/5 expressed as a decimal is 0.200 (or simply 0.2).

Question 4

Express the terminating decimals 0.875 and 0.375 each as vulgar fractions in their simplest form.

Accepted Answer

Introduction:This problem checks whether you can convert two terminating decimals, 0.875 and 0.375, into their equivalent fractional forms. Working comfortably between decimals and fractions is important in percentages, ratios, and many aptitude questions. Given Data / Assumptions: Decimals: 0.875 and 0.375. We must write each decimal as a fraction. Each fraction should be in simplest form. Concept / Approach:To convert a decimal to a fraction, write the decimal as an integer divided by a power of 10 (for example, 1000 if there are three decimal places) and then simplify. Terminating decimals always lead to fractions with denominators that involve powers of 2 and 5. We simplify by dividing numerator and denominator by their greatest common divisor. Step-by-Step Solution:Step 1: Convert 0.875 to a fraction. It has three decimal places, so write 0.875 = 875/1000.Step 2: Find the greatest common divisor of 875 and 1000. Both are divisible by 125: 875 = 125 * 7 and 1000 = 125 * 8.Step 3: Divide numerator and denominator by 125: 875/1000 = 7/8.Step 4: Convert 0.375 to a fraction: 0.375 = 375/1000.Step 5: Find the greatest common divisor of 375 and 1000. Both are divisible by 125: 375 = 125 * 3 and 1000 = 125 * 8.Step 6: Divide numerator and denominator by 125: 375/1000 = 3/8.Step 7: So 0.875 = 7/8 and 0.375 = 3/8. Verification / Alternative check:Convert back to decimals: 7/8 = 0.875 and 3/8 = 0.375. Both match the original decimals exactly, confirming that the conversions are correct and fully simplified. Why Other Options Are Wrong:The pairs 3/11 and 7/11, 7/9 and 9/13, and 7/9 and 9/11 give decimals that do not equal 0.875 or 0.375. The option 5/8 and 3/8 gives correct value for 0.375 (3/8) but 5/8 is 0.625, not 0.875. Only the pair 7/8 and 3/8 matches both decimals correctly. Common Pitfalls:Some learners simplify only partially or guess fractions without converting properly. Others may miscount decimal places and write, for example, 0.875 as 875/100 instead of 875/1000. Always match the number of decimal places to the correct power of 10 and then reduce the fraction by dividing by the greatest common divisor. Final Answer:The correct fractional forms are 0.875 = 7/8 and 0.375 = 3/8.

Question 5

When the decimal 0.46 is written as a fraction in simplest form, what is the sum of its numerator and denominator?

Accepted Answer

Introduction:This question combines decimal to fraction conversion with a small follow up calculation. Instead of asking directly for the simplified fraction, it asks for the sum of the numerator and denominator of that fraction, which is a common style in aptitude tests. Given Data / Assumptions: Decimal: 0.46. We must convert 0.46 to a fraction in simplest form. Then we must add the numerator and denominator of that fraction. Concept / Approach:To convert 0.46 to a fraction, note that there are two decimal places, so you can write 0.46 as 46/100. Then reduce the fraction by dividing numerator and denominator by their greatest common divisor. Once in lowest terms, simply sum the numerator and denominator. Step-by-Step Solution:Step 1: Write the decimal as a fraction: 0.46 = 46/100.Step 2: Find the greatest common divisor of 46 and 100.Step 3: Both 46 and 100 are divisible by 2: 46 = 2 * 23 and 100 = 2 * 50.Step 4: Divide numerator and denominator by 2: 46/100 = 23/50.Step 5: Check if 23 and 50 have any common factors greater than 1. They do not, since 23 is prime and does not divide 50.Step 6: So the simplest form is 23/50.Step 7: Compute the sum of numerator and denominator: 23 + 50 = 73. Verification / Alternative check:Convert 23/50 back to a decimal: 23 ÷ 50 = 0.46. This confirms that the simplified fraction is correct. The sum 73 therefore corresponds to the correct reduced fraction. Why Other Options Are Wrong:69, 96, 41, and 59 correspond to sums of other possible numerator and denominator pairs, but those fractions would not simplify to 0.46. Only 73 results from the correct fraction 23/50 that exactly equals 0.46. Common Pitfalls:A common mistake is to reduce 46/100 incorrectly, for example to 11.5/25 or to stop at 23/100. Others may attempt to simplify further by dividing numerator and denominator by 5 or 10 even when not possible. Carefully finding and applying the greatest common divisor avoids such errors. Final Answer:The sum of the numerator and denominator of the fraction representing 0.46 in simplest form is 73.

Question 6

If the notation 6.46 represents the recurring decimal 6.464646..., what is the correct fractional form of this recurring decimal?

Accepted Answer

Introduction:This question deals with converting a recurring decimal to a fraction. The notation 6.46 here is intended to represent the repeating decimal 6.464646..., where the block "46" repeats indefinitely. Converting recurring decimals to fractions is an important algebraic skill often tested in aptitude exams. Given Data / Assumptions: Recurring decimal: 6.464646..., where 46 repeats forever. We must express this number as a fraction. The answer should be in simplest fractional form. Concept / Approach:The standard method is to use algebra. Let x be the recurring decimal. Because two digits repeat, we multiply x by 100 to shift the decimal point two places to the right, creating a new equation that allows the repeating part to cancel when we subtract x from 100x. Step-by-Step Solution:Step 1: Let x = 6.464646...Step 2: Multiply both sides by 100 to move the decimal point two places: 100x = 646.464646...Step 3: Now subtract the original equation from this new equation: 100x − x = 646.464646... − 6.464646...Step 4: On the left side, 100x − x = 99x.Step 5: On the right side, the repeating decimals cancel, leaving 646 − 6 = 640.Step 6: So 99x = 640.Step 7: Solve for x: x = 640 / 99.Step 8: Therefore, the recurring decimal 6.464646... is equal to 640/99. Verification / Alternative check:To verify, divide 640 by 99. The result is approximately 6.46464646..., confirming that the fractional form matches the recurring decimal. No further simplification is possible because 640 and 99 have no common factor greater than 1. Why Other Options Are Wrong:640/90 simplifies to 64/9, which is approximately 7.11, not 6.46 recurring. 64000/99 is far too large and corresponds to about 646.4646.... 640/9 is around 71.11. 646/99 is about 6.525..., not equal to 6.4646.... Only 640/99 reproduces the exact recurring decimal 6.464646.... Common Pitfalls:Some learners mistakenly treat 6.46 as a terminating decimal 6.46 exactly, not recognizing the repeating pattern intended. Others may forget to subtract x from 100x correctly or mis-handle the decimal alignment. Carefully setting up the algebraic equations and subtracting to eliminate the repeating part is the key technique. Final Answer:The correct fractional representation of the recurring decimal 6.464646... is 640/99.

Question 7

It is known that 52,416 divided by 312 gives a quotient of 168. Using this information, what will be the quotient when 52.416 is divided by 0.0168?

Accepted Answer

Introduction:This problem tests your understanding of how scaling both the dividend and the divisor by powers of 10 affects the quotient. Instead of directly performing a messy decimal division, you can use the given exact division involving whole numbers to find the new quotient efficiently. Given Data / Assumptions: Given: 52,416 ÷ 312 = 168. We must find: 52.416 ÷ 0.0168. The relationship between 52,416 and 52.416, and between 312 and 0.0168, involves powers of 10. Concept / Approach:When both the numerator (dividend) and the denominator (divisor) of a division are multiplied or divided by the same nonzero number, the quotient does not change. By expressing 52.416 and 0.0168 in terms of 52,416 and 168/10,000, we can rewrite the division to leverage the known result 52,416 ÷ 312 = 168. Step-by-Step Solution:Step 1: Express 52.416 as a fraction of 52,416: 52.416 = 52,416 / 1000.Step 2: Express 0.0168 as a fraction: 0.0168 = 168 / 10,000.Step 3: Write the required division: 52.416 ÷ 0.0168 = (52,416 / 1000) ÷ (168 / 10,000).Step 4: Dividing by a fraction is the same as multiplying by its reciprocal: (52,416 / 1000) * (10,000 / 168).Step 5: Simplify: (52,416 * 10,000) / (1000 * 168) = (52,416 * 10) / 168.Step 6: So the result is 52,4160 / 168.Step 7: Note that 52,416 ÷ 312 = 168 is given. We can see that 52,4160 = 10 * 52,416.Step 8: Compute 52,4160 / 168 = 10 * (52,416 / 168).Step 9: Since 52,416 / 312 = 168, and 168 * 312 = 52,416, we can verify that 52,416 / 168 = 312.Step 10: Therefore, 52,4160 / 168 = 10 * 312 = 3120.Thus, the desired quotient is 3120. Verification / Alternative check:We can quickly approximate: 52.416 ÷ 0.0168 is like 52 ÷ 0.017, which is roughly 3,000. The exact answer 3120 is close to this estimate and matches the detailed calculation. Why Other Options Are Wrong:Values such as 32.10, 321.0, 3.120, and 0.312 are far too small compared to the approximate ratio of 52 to 0.017. They arise from misplacing decimal points or dividing incorrectly by powers of 10. The only value consistent with the scaling and known division fact is 3120. Common Pitfalls:Many students attempt to do the decimal division directly, leading to errors with decimal placement. Others may incorrectly cancel zeros without keeping track of the effect on both numerator and denominator. Using the relationship between the given whole number division and the target division simplifies the problem and avoids such mistakes. Final Answer:The quotient when 52.416 is divided by 0.0168 is 3120.

Question 8

The decimal number 0.127 is how much greater than the fraction 1/8?

Accepted Answer

Introduction:This question measures your skill in comparing a decimal with a fraction and finding their difference as a fraction. It is a typical type of problem in decimal–fraction conversion and comparison topics in quantitative aptitude. Given Data / Assumptions: Decimal: 0.127. Fraction: 1/8. We must compute how much greater 0.127 is than 1/8. The final answer should be expressed as a fraction. Concept / Approach:To find how much greater one number is than another, we subtract the smaller from the larger. So we calculate 0.127 − 1/8. It is convenient to convert both numbers to decimals or both to fractions. Since 0.127 is already a decimal, we can convert it to a fraction and subtract 1/8 in fractional form. Step-by-Step Solution:Step 1: Convert 1/8 to a decimal or leave it as a fraction. 1/8 = 0.125.Step 2: Compare 0.127 and 0.125; clearly, 0.127 is larger.Step 3: Compute the decimal difference: 0.127 − 0.125 = 0.002.Step 4: Now express 0.002 as a fraction. 0.002 = 2/1000.Step 5: Simplify 2/1000 by dividing numerator and denominator by 2: 2/1000 = 1/500.Step 6: Therefore, 0.127 is greater than 1/8 by 1/500. Verification / Alternative check:Convert 1/500 back to decimal: 1 ÷ 500 = 0.002. Note that 1/8 = 0.125. Adding 0.002 to 0.125 gives 0.127, which confirms that the difference is indeed 1/500. Why Other Options Are Wrong:1/100 equals 0.01, which is too large. 1/1500 equals approximately 0.000666..., much smaller than 0.002. 1/2 equals 0.5, clearly unrelated to this small difference. 1/200 equals 0.005, also too large. Only 1/500 corresponds to the precise difference of 0.002 between 0.127 and 0.125. Common Pitfalls:Students sometimes subtract 1/8 from 0.127 incorrectly due to decimal misalignment or confuse which number is larger. Others may convert the fraction to an imprecise decimal and then round too early. Keeping a few decimal places and doing exact subtraction, or using fraction forms from the start, ensures accuracy. Final Answer:The number 0.127 is greater than 1/8 by 1/500.

Question 9

Compute the value of 25.7322 − 15.7322. What is the exact result of this subtraction?

Accepted Answer

Introduction:This is a straightforward problem involving subtraction of decimal numbers. Such questions test your ability to align decimal points correctly and subtract without making place value mistakes, which is an essential arithmetic skill. Given Data / Assumptions: We must evaluate: 25.7322 − 15.7322. Both numbers have four decimal places. We are asked for the exact result. Concept / Approach:When subtracting decimals, the key is to align the decimal points so that digits of the same place value (ones, tenths, hundredths, thousandths, ten-thousandths) line up vertically. Then, subtract column by column just as with whole numbers. Because the decimal parts in this problem are identical, the subtraction simplifies nicely. Step-by-Step Solution:Step 1: Write the numbers aligned by decimal point:25.732215.7322Step 2: Subtract the decimal parts. Since 0.7322 − 0.7322 = 0, the decimal portion cancels out entirely.Step 3: Now subtract the whole number parts: 25 − 15 = 10.Step 4: Combine the results. There is no remaining decimal part, so the final answer is exactly 10.0000, which we usually write as 10. Verification / Alternative check:We can also see that 25.7322 − 15.7322 = (15.7322 + 10) − 15.7322. The 15.7322 terms cancel, leaving 10, which confirms the earlier calculation. Another quick check is to think of the subtraction as removing the same fractional part from both numbers, leaving only the difference in whole numbers. Why Other Options Are Wrong:9.64, 11.64, 10.5, and 12 do not match the exact difference. Any decimal result would imply that the decimal parts were not identical. Since the digits after the decimal point are exactly the same in both numbers, the subtraction must result in a whole number. Only 10 satisfies this reasoning and matches the detailed calculation. Common Pitfalls:Common mistakes include misaligning the decimal points, subtracting the wrong digits, or incorrectly assuming a non zero decimal result because of the presence of decimal digits. Remember that if two numbers have identical decimal parts, subtracting them will cancel the decimal, leaving the difference of the whole number parts alone. Final Answer:The exact value of 25.7322 − 15.7322 is 10.

Question 10

What is the decimal representation of the fraction 1/8 (one divided by eight)?

Accepted Answer

Introduction:This question focuses on converting the simple fraction 1/8 into its decimal form. Understanding how to convert fractions to decimals is crucial for dealing with percentages, ratios, and many practical calculations in aptitude exams and daily life. Given Data / Assumptions: Fraction: 1/8. We must express this as a decimal. The answer should be exact, not rounded. Concept / Approach:To convert a fraction a/b into a decimal, you perform the division a ÷ b. Because 8 is a power of 2, the decimal will terminate. It is useful to know that dividing 1 by a power of 2 gives a neat terminating decimal. Step-by-Step Solution:Step 1: Interpret 1/8 as the division 1 ÷ 8.Step 2: Since 1 is less than 8, consider 1.000 and divide by 8.Step 3: 8 goes into 10 once (1 * 8 = 8), leaving a remainder of 2. Bring down the next 0 to get 20.Step 4: 8 goes into 20 twice (2 * 8 = 16), leaving a remainder of 4. Bring down the next 0 to get 40.Step 5: 8 goes into 40 five times (5 * 8 = 40), leaving a remainder of 0.Step 6: So the quotient is 0.125.Thus, 1/8 expressed as a decimal is 0.125. Verification / Alternative check:We can also reason that 1/8 = (0.5) / 4 because 1/2 = 0.5 and 1/8 = 1 / (2^3). Dividing 0.5 by 4 gives 0.125, which is consistent with our long division result. Why Other Options Are Wrong:0.375 is the decimal for 3/8, not 1/8. 0.247 and 0.214 are unrelated decimals. 0.200 is equal to 1/5, not 1/8. Only 0.125 exactly equals 1/8. Common Pitfalls:Some students confuse 1/8 with 1/4 (0.25) or 3/8 (0.375). Another common mistake is performing the division quickly and misplacing the decimal. Carefully carrying out long division or remembering common fraction–decimal equivalences helps avoid such errors. Final Answer:The decimal representation of 1/8 is 0.125.

Question 11

Express the fractions 3/8 and 1/10 as decimals. What are their decimal representations?

Accepted Answer

Introduction:This question evaluates your ability to convert two simple fractions, 3/8 and 1/10, into decimal form. Such conversions are used heavily in percentage calculations, ratio problems, and financial mathematics. Given Data / Assumptions: Fractions: 3/8 and 1/10. We must express each as a decimal. The answers should be exact, terminating decimals. Concept / Approach:For each fraction a/b, we convert to decimal by performing the division a ÷ b. Since 8 and 10 are composed of factors 2 and 5, both 3/8 and 1/10 have terminating decimal expansions. We can compute them directly or recall well-known fraction–decimal pairs. Step-by-Step Solution:Step 1: Convert 3/8 to a decimal: 3 ÷ 8.Step 2: Perform the division: 8 goes into 30 three times (3 * 8 = 24), remainder 6; bring down a 0 to make 60.Step 3: 8 goes into 60 seven times (7 * 8 = 56), remainder 4; bring down a 0 to make 40.Step 4: 8 goes into 40 five times (5 * 8 = 40), remainder 0.Step 5: The quotient is 0.375, so 3/8 = 0.375.Step 6: Convert 1/10 to a decimal: 1 ÷ 10 = 0.1, because moving the decimal point one place left divides by 10.Step 7: Therefore, the decimal representations are 0.375 and 0.1. Verification / Alternative check:Write them as fractions with denominators that are powers of 10. For 3/8, multiply numerator and denominator by 125: 3/8 = (3 * 125)/(8 * 125) = 375/1000 = 0.375. For 1/10, it is already one tenth, so directly 0.1. These match our division results exactly. Why Other Options Are Wrong:0.375 and 1 implies 1/10 equals 1, which is incorrect. 1.375, 13.75, and 1.1 are far from the true values and come from misplacing decimal points. 0.3 and 0.1 would correspond to 3/10 and 1/10, not 3/8 and 1/10. Only the pair 0.375 and 0.1 correctly represents 3/8 and 1/10. Common Pitfalls:Students sometimes mix up 3/8 with 3/10 or 1/8, or incorrectly perform the division 3 ÷ 8, leading to wrong decimals. Another error is to assume 1/10 is 0.01 rather than 0.1. Remember that dividing by 10 shifts the decimal one place to the left, not two places. Final Answer:The decimal forms are 3/8 = 0.375 and 1/10 = 0.1.

Question 12

What is the value of the fraction 3/4 divided by 2, expressed again as a simplified fraction?

Accepted Answer

Introduction:This question involves division of a fraction by a whole number. It checks whether you understand how to handle division of fractions and can reduce the result to simplest form. Such operations appear frequently in ratio, proportion, and algebra problems. Given Data / Assumptions: We are given the fraction 3/4. We must divide 3/4 by 2. The answer should be given as a simplified fraction. Concept / Approach:Dividing a fraction by a whole number is equivalent to multiplying the fraction by the reciprocal of that whole number. In general, (a/b) ÷ c = (a/b) * (1/c) = a/(b * c). After performing the multiplication, we reduce the resulting fraction if possible. Step-by-Step Solution:Step 1: Start with the expression: (3/4) ÷ 2.Step 2: Write 2 as a fraction: 2 = 2/1.Step 3: Division by a fraction is multiplication by its reciprocal: (3/4) ÷ (2/1) = (3/4) * (1/2).Step 4: Multiply the numerators and denominators: (3 * 1) / (4 * 2) = 3 / 8.Step 5: Check if 3/8 can be simplified. Since 3 and 8 have no common factor greater than 1, 3/8 is already in simplest form. Verification / Alternative check:As a decimal, 3/4 = 0.75. Dividing 0.75 by 2 gives 0.375. Converting 3/8 to decimal also gives 0.375. Since both methods produce the same decimal, this confirms that 3/8 is the correct fraction for (3/4) ÷ 2. Why Other Options Are Wrong:3/2 is larger than 1 and would result from multiplying 3/4 by 2, not dividing. 2/3 and 8/3 come from unrelated operations. 1/4 is 0.25 and does not match 0.375. Only 3/8 corresponds to the correct result of dividing 3/4 by 2. Common Pitfalls:Students sometimes mistakenly multiply by 2 instead of dividing or forget to take the reciprocal when handling fraction division. Another mistake is to divide only the numerator or only the denominator by 2 instead of using the proper method. Remember that dividing by a whole number c is equivalent to multiplying by 1/c. Final Answer:The value of 3/4 divided by 2 is 3/8.

Question 13

When the fraction 3/8 is converted to a decimal number, what is its value?

Accepted Answer

Introduction:This problem simply asks you to convert the fraction 3/8 into its decimal equivalent. Mastering fraction to decimal conversion is essential for working with percentages, ratios, and many other numerical questions in aptitude tests. Given Data / Assumptions: Fraction: 3/8. We must express it as a decimal. We want the exact terminating decimal. Concept / Approach:To convert a fraction to a decimal, perform the division of numerator by denominator. Because 8 is a power of 2, we know the decimal will terminate. Alternatively, we can express 3/8 with a denominator that is a power of 10 and then read off the decimal representation. Step-by-Step Solution:Step 1: Interpret 3/8 as 3 ÷ 8.Step 2: Perform the division using long division or mental math. 8 does not go into 3, so consider 30 tenths.Step 3: 8 goes into 30 three times (3 * 8 = 24), remainder 6. Bring down a 0 to make 60.Step 4: 8 goes into 60 seven times (7 * 8 = 56), remainder 4. Bring down a 0 to make 40.Step 5: 8 goes into 40 five times (5 * 8 = 40), remainder 0.Step 6: Therefore, the quotient is 0.375. Verification / Alternative check:We can express 3/8 with denominator 1000: multiply numerator and denominator by 125: (3 * 125)/(8 * 125) = 375/1000. Writing 375/1000 as a decimal gives 0.375, confirming our long division result. Why Other Options Are Wrong:1.375 and 2.375 are larger than 1 and would require a numerator greater than the denominator. 3.75 is ten times the correct value and corresponds to 30/8, not 3/8. 0.735 is just a rearrangement of the digits and does not match the correct decimal of 3/8. Only 0.375 equals 3/8 exactly. Common Pitfalls:Students may misplace the decimal point or stop the division too early, leading to partial or incorrect results like 0.37. Others confuse 3/8 with 3/10 or 1/8. Knowing a few common equivalents, such as 1/8 = 0.125 and 3/8 = 0.375, can save time. Final Answer:The decimal equivalent of 3/8 is 0.375.

Question 14

Which of the following fractions has the greatest value: 7/12, 9/16, 13/20 or 32/80?

Accepted Answer

Introduction:This question asks you to compare several fractions and determine which one is the largest. Comparing fractions is a fundamental skill in arithmetic and is often required in data interpretation and percentage problems. Given Data / Assumptions: Fractions to compare: 7/12, 9/16, 13/20, and 32/80. We must identify the fraction with the greatest numerical value. All fractions are positive. Concept / Approach:There are two main approaches: convert each fraction to a decimal or convert them to a common denominator. Because the denominators differ, converting to decimals is often quicker for multiple choice questions. Then we simply compare the decimal values to see which is the largest. Step-by-Step Solution:Step 1: Convert 7/12 to a decimal: 7 ÷ 12 ≈ 0.5833.Step 2: Convert 9/16 to a decimal: 9 ÷ 16 = 0.5625.Step 3: Convert 13/20 to a decimal: 13 ÷ 20 = 0.65.Step 4: Convert 32/80 to a decimal: first simplify 32/80 by dividing numerator and denominator by 16 to get 2/5; 2/5 = 0.4.Step 5: Now list the decimals: 7/12 ≈ 0.5833, 9/16 = 0.5625, 13/20 = 0.65, 32/80 = 0.4.Step 6: Compare: 0.65 is larger than 0.5833, 0.5625, and 0.4.Step 7: Therefore, 13/20 is the largest fraction. Verification / Alternative check:We can also compare by cross multiplication. For example, compare 13/20 and 7/12 by cross multiplying: 13 * 12 = 156 and 7 * 20 = 140. Since 156 > 140, 13/20 > 7/12. Similarly, compare 13/20 with 9/16: 13 * 16 = 208 and 9 * 20 = 180, so 13/20 > 9/16. Since 32/80 is only 2/5, it is clearly smaller than 13/20 which is 0.65. This confirms that 13/20 is the greatest. Why Other Options Are Wrong:7/12 and 9/16 are both less than 2/3 and have decimals below 0.6. 32/80 simplifies to 2/5, which is 0.4, much smaller than 0.65. The distractor 17/30 is about 0.5667 and is not even among the original fractions to compare. Only 13/20 has the highest decimal value among the listed fractions. Common Pitfalls:Students sometimes compare only numerators or only denominators without considering both together. Another mistake is miscalculating decimals or simplifying incorrectly, especially with 32/80. Always either simplify first or compute accurate decimals to compare fractions reliably. Final Answer:Among the given options, the largest fraction is 13/20.

Question 15

In basic decimal fraction conversion, what is the value of the fraction 3/5 when it is expressed as a decimal number?

Accepted Answer

Introduction / Context: Decimal fractions appear very frequently in aptitude exams, and you are often required to convert between fractions and decimals quickly. In this question, we are asked to convert the rational number 3/5 into its decimal form. Understanding how to perform this basic conversion helps in percentage problems, ratio comparisons, and many data interpretation questions. Given Data / Assumptions: We are given the fraction 3/5. We need to express 3/5 as a decimal number. No rounding to fewer decimal places is strictly required, but the options are given with two decimal places. Concept / Approach: The key idea is that any fraction a/b can be written as a decimal by performing the division a ÷ b. Here, the fraction 3/5 can be turned into a decimal by dividing 3 by 5. Another useful approach is to convert the denominator to 10, 100, or 1000, because decimals are based on powers of 10. Once the denominator is turned into 10, the numerator directly indicates the decimal value. Step-by-Step Solution: Step 1: Start with the given fraction 3/5. Step 2: Notice that 5 can be converted into 10 by multiplying by 2. Step 3: Multiply both numerator and denominator by 2: (3 * 2) / (5 * 2) = 6/10. Step 4: The fraction 6/10 is equal to the decimal 0.6. Step 5: Writing 0.6 with two decimal places gives 0.60, which matches one of the answer options. Verification / Alternative Check: Another way is to perform direct long division: 3 ÷ 5. Since 5 goes into 3 zero times, write 0.something and consider 30 ÷ 5. Then 5 goes into 30 exactly 6 times, giving 0.6. Writing it as 0.60 does not change its value; it is still six tenths. This confirms that 3/5 = 0.60 in decimal notation. Why Other Options Are Wrong: 0.25 corresponds to 1/4, not 3/5. 0.45 equals 45/100 or 9/20, which is less than 3/5. 0.75 is equal to 3/4, which is greater than 3/5. Therefore, these values do not represent 3/5. Common Pitfalls: Students sometimes confuse 3/5 with common benchmark fractions like 1/2 (0.5) or 3/4 (0.75). Another mistake is to think that 3 divided by 5 must be greater than 1, which is incorrect because the denominator is larger than the numerator. Remember that if the numerator is smaller than the denominator, the decimal will be less than 1. Final Answer: The correct decimal representation of 3/5 is 0.60.

Question 16

For the fractions 5/8, 7/12, 3/4 and 13/16, which of the following options shows them correctly arranged in ascending order of their numerical value?

Accepted Answer

Introduction / Context: Comparing and ordering fractions is a fundamental arithmetic skill that is very important in aptitude exams. In this problem, we are given four fractions and asked to choose the option that arranges them in ascending order, that is, from the smallest value to the largest value. Being comfortable with fraction comparison helps in topics like ratios, percentages and data interpretation. Given Data / Assumptions: Fractions given: 5/8, 7/12, 3/4 and 13/16. We must arrange them in increasing order (ascending order). All fractions are positive proper fractions (less than 1). Concept / Approach: There are several ways to compare fractions: Convert each fraction to an equivalent fraction with a common denominator. Convert each fraction to a decimal value and then compare the decimals. Use cross-multiplication to compare pairs of fractions. For explanation clarity, converting to decimal or using approximate values works very well in exam conditions. Step-by-Step Solution: Step 1: Convert 7/12 to a decimal: 7 ÷ 12 ≈ 0.5833. Step 2: Convert 5/8 to a decimal: 5 ÷ 8 = 0.625. Step 3: Convert 3/4 to a decimal: 3 ÷ 4 = 0.75. Step 4: Convert 13/16 to a decimal: 13 ÷ 16 = 0.8125. Step 5: Now compare these decimal values: 0.5833, 0.625, 0.75, 0.8125. Step 6: Ordering them from smallest to largest gives 0.5833 < 0.625 < 0.75 < 0.8125. Step 7: Translating back to fractions, this order is 7/12 < 5/8 < 3/4 < 13/16. Verification / Alternative Check: We can also check using cross-multiplication. For example, to compare 7/12 and 5/8, compare 7 * 8 = 56 and 5 * 12 = 60. Since 56 < 60, we have 7/12 < 5/8. Similar pairwise checks will confirm the order 7/12 < 5/8 < 3/4 < 13/16. Why Other Options Are Wrong: Option A: 5/8 < 7/12 is incorrect because 7/12 is actually smaller than 5/8. Option B: Again starts with 5/8 and 7/12 in the wrong order and also misplaces 13/16 and 3/4. Option C: Puts 7/12 last, which is wrong since 7/12 is the smallest fraction. Only option D shows the correct ascending order. Common Pitfalls: A common error is to compare only numerators or only denominators without considering the fraction as a whole. Another mistake is to think that the larger denominator always gives a smaller fraction, which is not correct unless numerators are equal. Always compare fractions logically, either by using a common denominator, decimals, or cross-multiplication. Final Answer: The correct ascending order is 7/12 < 5/8 < 3/4 < 13/16.

Question 17

In decimal notation, what is the value of the fraction 2/3 when it is expressed correct to two decimal places?

Accepted Answer

Introduction / Context: Converting rational numbers (fractions) to decimal form is a basic but essential skill in quantitative aptitude. In this question, we are asked to convert the fraction 2/3 into a decimal and then match it with the closest choice given, expressed using two decimal places. This type of conversion is also frequently used in percentage calculations and comparisons in competitive exams. Given Data / Assumptions: Given fraction: 2/3. We need its decimal representation. Options use two decimal places, so rounding or truncation must be considered. Concept / Approach: To convert a fraction to a decimal, divide the numerator by the denominator. Here, we need to compute 2 ÷ 3. The result will be a repeating decimal. Once we know the repeating decimal, we can round it to two decimal places to match one of the answer choices. Remember basic rules of rounding: if the third decimal place is 5 or more, we round the second place up by 1. Step-by-Step Solution: Step 1: Start with the fraction 2/3. Step 2: Perform the division 2 ÷ 3. Step 3: 3 goes into 2 zero times, so we write 0.x and proceed with 20 ÷ 3. Step 4: 3 goes into 20 six times (6 * 3 = 18) with remainder 2, giving 0.6. Step 5: The remainder 2 repeats, so the digits 6 continue: 2/3 = 0.6666..., a recurring decimal. Step 6: To express 0.6666... to two decimal places, we look at the third decimal digit: 0.666... has 6 as the third digit. Step 7: Because the third digit (6) is greater than or equal to 5, we round 0.66 up to 0.67. Verification / Alternative Check: We know that 1/3 = 0.3333..., and multiplying both sides by 2 gives 2/3 = 0.6666.... Again, rounding 0.6666... to two decimal places yields 0.67, so this confirms our result through a simple proportional argument. Why Other Options Are Wrong: 0.47 is far smaller than 2/3 and does not approximate 0.6666.... 6.77 is much larger than 1, whereas 2/3 is less than 1, so this option is absurd here. 0.07 is extremely small and clearly not equal to 2/3. Only 0.67 is a correct two decimal place approximation to 2/3. Common Pitfalls: Learners sometimes confuse 2/3 with 0.63 or 0.6 due to careless rounding or truncation. Truncating 0.6666... to 0.66 instead of rounding it to 0.67 is a frequent error. Always use proper rounding rules and ensure you understand the difference between truncation and rounding. Final Answer: The decimal representation of 2/3 correct to two decimal places is 0.67.

Question 18

Among the following rational numbers, which one is a reducible fraction, that is, not already in its lowest terms?

Accepted Answer

Introduction / Context: Fractions can often be simplified by dividing the numerator and the denominator by their greatest common divisor. A fraction that can be simplified is called a reducible fraction, while one that cannot be reduced further is said to be in lowest terms. This question tests your ability to recognize which of the given fractions can still be simplified. Given Data / Assumptions: Given fractions: 79/26, 105/112, 41/17 and 91/15. We must identify which fraction is reducible (not in lowest terms). All numbers are positive. Concept / Approach: A fraction a/b is reducible if the numerator and denominator have a common factor greater than 1. To check this, we find the greatest common divisor (GCD) or simply test divisibility by small primes such as 2, 3, 5, 7, 11, and so on. If no common factor is found, the fraction is already in simplest form. Step-by-Step Solution: Step 1: Consider 79/26. 79 is a prime number and 26 = 2 * 13. There is no common factor, so 79/26 is irreducible. Step 2: Consider 105/112. Factor 105: 105 = 3 * 5 * 7. Factor 112: 112 = 2 * 2 * 2 * 2 * 7. Step 3: Both numbers share a common factor 7, so we can simplify. Step 4: Divide numerator and denominator by 7: (105 ÷ 7) / (112 ÷ 7) = 15/16. Step 5: Since 105/112 reduces to 15/16, it is definitely a reducible fraction. Step 6: Check 41/17: 41 is prime and 17 is prime, so there is no common factor. It is in lowest terms. Step 7: Check 91/15: 91 = 7 * 13 and 15 = 3 * 5. No common factors exist, so it is also in lowest terms. Verification / Alternative Check: We could also quickly test each fraction by trying to divide both numerator and denominator by small primes. Only 105 and 112 are divisible by 7, confirming that 105/112 simplifies to 15/16. This is a fast mental method in exams. Why Other Options Are Wrong: 79/26 has no common factor other than 1 between 79 and 26, so it is already simplified. 41/17 uses two prime numbers with no common divisor, so it cannot be reduced. 91/15 has factors 7 * 13 and 3 * 5, which again share no common divisor. Therefore, these three fractions are not reducible, leaving only 105/112 as the correct choice. Common Pitfalls: One common mistake is to assume that any fraction with large numbers must be reducible. Another is to miss common factors like 7 or 11 because students usually check only 2, 3 and 5. Always factor or at least test divisibility by a few primes when time permits. Final Answer: The fraction that is reducible is 105/112.

Question 19

Let x be a non-zero number such that the sum of the number and four times its reciprocal is 17/2. Which of the following is a possible value of x?

Accepted Answer

Introduction / Context: This problem involves a simple algebraic equation that connects a number with its reciprocal. Such questions test your ability to translate a verbal condition into an algebraic expression and then solve the resulting quadratic equation. Recognizing patterns in expressions that involve a variable and its reciprocal is very useful in many aptitude and algebra questions. Given Data / Assumptions: x is a non-zero real number. The relationship given is: x + 4 * (1/x) = 17/2. We must determine which option for x satisfies this equation. Concept / Approach: To solve equations involving a number and its reciprocal, we multiply through by the variable to eliminate the denominator. This usually leads to a quadratic equation in standard form a*x^2 + b*x + c = 0. We can then solve this quadratic using factorization or the quadratic formula. After finding the roots, we check them against the given options and discard any that are not listed. Step-by-Step Solution: Step 1: Start from the equation: x + 4/x = 17/2. Step 2: Multiply both sides by 2x to remove the denominator: 2x * x + 2x * (4/x) = 2x * 17/2. Step 3: Simplify terms: 2x^2 + 8 = 17x. Step 4: Rearrange into standard quadratic form: 2x^2 - 17x + 8 = 0. Step 5: Use the quadratic formula: x = [17 ± sqrt(17^2 - 4 * 2 * 8)] / (2 * 2). Step 6: Compute the discriminant: 17^2 = 289; 4 * 2 * 8 = 64; so discriminant = 289 - 64 = 225. Step 7: The square root of 225 is 15, so x = (17 ± 15) / 4. Step 8: This gives x = (17 + 15) / 4 = 32/4 = 8 or x = (17 - 15) / 4 = 2/4 = 1/2. Verification / Alternative Check: Check x = 8 in the original equation: 8 + 4 * (1/8) = 8 + 4/8 = 8 + 1/2 = 8.5 = 17/2, so x = 8 satisfies the condition. The other solution, x = 1/2, also satisfies the equation but it does not appear among the options. Therefore, the only allowable value from the options is 8. Why Other Options Are Wrong: If x = 4, then 4 + 4 * (1/4) = 4 + 1 = 5, which is not 17/2. If x = 12, then 12 + 4 * (1/12) = 12 + 1/3 ≈ 12.33, not 8.5. If x = 16, then 16 + 4 * (1/16) = 16 + 1/4 = 16.25, again not 17/2. Thus these options do not satisfy the given equation. Common Pitfalls: Students sometimes forget to multiply by x correctly and make sign errors when bringing all terms to one side of the equation. Another common mistake is to ignore the second root of the quadratic. Even though 1/2 is a correct mathematical solution, you must still check the answer choices and select the root that is actually listed. Final Answer: A value of x that satisfies the condition and appears in the options is 8.

Question 20

A fraction is greater than its reciprocal by 9/20. What is the value of this fraction?

Accepted Answer

Introduction / Context: This question uses a relationship between a fraction and its reciprocal. Problems of this type commonly appear in algebra sections of aptitude tests. They test your ability to form linear or quadratic equations from verbal statements and then solve them accurately. Here, the fraction is greater than its reciprocal by a fixed amount, 9/20. Given Data / Assumptions: Let the fraction be x (x ≠ 0). Its reciprocal is 1/x. The fraction is greater than its reciprocal by 9/20, so x - (1/x) = 9/20. We need to find the value of x from the given options. Concept / Approach: The verbal statement is converted into an algebraic equation involving x and 1/x. To eliminate the denominator, we multiply the entire equation by x, giving a quadratic equation. Quadratic equations of the form a*x^2 + b*x + c = 0 can be solved using the quadratic formula. Finally, we match the valid root with the options provided, typically choosing the positive fraction that fits the context. Step-by-Step Solution: Step 1: Write the equation based on the statement: x - 1/x = 9/20. Step 2: Multiply both sides by x to remove the denominator: x * x - 1 = (9/20) * x. Step 3: Simplify to get: x^2 - 1 = (9/20)x. Step 4: Multiply through by 20 to clear the fraction: 20x^2 - 20 = 9x. Step 5: Rearrange into standard quadratic form: 20x^2 - 9x - 20 = 0. Step 6: Use the quadratic formula: x = [9 ± sqrt(9^2 - 4 * 20 * (-20))] / (2 * 20). Step 7: Compute the discriminant: 9^2 = 81; 4 * 20 * 20 = 1600; so 81 + 1600 = 1681. Step 8: The square root of 1681 is 41, so x = (9 ± 41) / 40. Step 9: The roots are x = (9 + 41) / 40 = 50/40 = 5/4 and x = (9 - 41) / 40 = -32/40 = -4/5. Verification / Alternative Check: Test x = 5/4 in the original equation: x - 1/x = 5/4 - 4/5. Compute 5/4 - 4/5 = (25/20) - (16/20) = 9/20, which matches the condition. The second root, -4/5, also satisfies the equation algebraically but is negative. The options provided include 5/4 and 4/5, not -4/5, so we select the positive fraction 5/4. Why Other Options Are Wrong: 4/5 would give 4/5 - 5/4, which is negative, not 9/20. 3/4 and 4/3 also do not satisfy the given equation when substituted into x - 1/x. Their differences from their reciprocals will be different values, not 9/20. Common Pitfalls: Students sometimes forget that both positive and negative roots can arise when solving quadratics. Another mistake is to invert the relationship and write 1/x - x instead of x - 1/x. Always read the phrase carefully: “fraction is greater than its reciprocal by” means fraction minus reciprocal equals the given positive difference. Final Answer: The required fraction is 5/4.

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