Q: When the repeating decimal 0.090909... is written as a fraction in simplest form, what is that fraction?

Introduction / Context:This question tests your understanding of how to convert a repeating decimal into an exact fraction. Repeating decimals often appear in algebra and number system problems, and knowing how to handle them shows deeper conceptual understanding.Given Data / Assumptions:Repeating decimal: 0.090909..., where 09 repeats indefinitely.We must express this as a fraction in simplest form.The dots indicate that the pattern 09 continues without end.Concept / Approach:To convert a repeating decimal into a fraction, we use an algebraic method. We denote the decimal by a variable, multiply by a suitable power of 10 so that the repeating part lines up, and then subtract to eliminate the repetition.Step-by-Step Solution:Step 1: Let x = 0.090909..., with 09 repeating.Step 2: Since the repeating block has two digits, multiply both sides by 100: 100x = 9.090909...Step 3: Now subtract the original x from this new equation: 100x - x = 9.090909... - 0.090909....Step 4: On the right-hand side, the repeating parts cancel, leaving 9.Step 5: So 99x = 9, which implies x = 9/99.Step 6: Simplify 9/99 by dividing numerator and denominator by 9 to get 1/11.Verification / Alternative check:We know that 1/11 is 0.090909..., so converting back to a decimal immediately confirms that our fraction form is correct.Why Other Options Are Wrong:1/33 equals approximately 0.030303..., which has a different repeating block. 2/33 equals about 0.060606..., also incorrect. 6/11 equals about 0.545454..., which is much larger and contains 54 as the repeating pair, not 09.Common Pitfalls:A common error is misidentifying the length of the repeating block or multiplying by 10 instead of 100 in this case. Another mistake is simplifying 9/99 incorrectly or not simplifying it at all, forgetting to reduce the fraction to lowest terms.Final Answer:The repeating decimal 0.090909... is equal to the fraction 1/11.

Q: The sum of a fraction and ten times its reciprocal is 37/4. What is the fraction?

Introduction / Context:This problem again uses a relationship between a fraction and its reciprocal, with a larger coefficient on the reciprocal term. It is an algebraic question that reinforces skills in forming and solving quadratic equations.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.The reciprocal is 1/x.The condition is x + 10 * (1/x) = 37/4.We need to find x among the given options.Concept / Approach:We translate the condition into an equation, clear denominators, and solve the quadratic equation that results. Finally, we see which algebraic solution appears in the answer choices.Step-by-Step Solution:Step 1: Write the equation: x + 10/x = 37/4.Step 2: Multiply both sides by 4x to clear fractions: 4x * x + 4x * (10/x) = 4x * (37/4).Step 3: Simplify to get 4x^2 + 40 = 37x.Step 4: Rearrange into standard form: 4x^2 - 37x + 40 = 0.Step 5: Compute the discriminant: D = (-37)^2 - 4 * 4 * 40 = 1369 - 640 = 729. The square root of 729 is 27.Step 6: Solve for x: x = [37 ± 27] / (2 * 4) = (37 ± 27) / 8. Thus x = 64/8 = 8 or x = 10/8 = 5/4.Verification / Alternative check:Check x = 5/4: 5/4 + 10 * (4/5) = 5/4 + 8 = 5/4 + 32/4 = 37/4, which matches the given sum. The value x = 8 also satisfies the equation, because 8 + 10 * (1/8) = 8 + 1.25 = 9.25 = 37/4. However, 8 is not listed among the options, whereas 5/4 is.Why Other Options Are Wrong:Substituting 4/5, 3/4 or 4/3 into x + 10/x does not produce 37/4. Their resulting sums are quite different, so these options do not satisfy the condition.Common Pitfalls:Mistakes can occur when clearing denominators and simplifying the quadratic. Some candidates may forget that there can be two valid algebraic solutions but only one matching the provided options. Always test candidate values back in the original equation.Final Answer:The fraction that fits the condition and appears in the choices is 5/4.

Q: When the repeating decimal 0.151515... is written as a fraction in lowest terms, what is that fraction?

Introduction / Context:This question focuses on converting a repeating decimal with a two-digit repeating block into a fraction. Understanding this method strengthens algebraic manipulation skills and number sense.Given Data / Assumptions:Repeating decimal: 0.151515..., with 15 repeating indefinitely.We must express this decimal as a fraction in simplest form.The repeating pattern is constant throughout the decimal part.Concept / Approach:The standard technique is to assign the decimal to a variable, multiply by a power of 10 equal to the length of the repeating block, and then subtract to eliminate the repeating part. The remaining equation is easy to convert into a fraction.Step-by-Step Solution:Step 1: Let x = 0.151515..., where 15 repeats.Step 2: Since the repeating block has two digits, multiply both sides by 100: 100x = 15.151515...Step 3: Subtract the original equation from this new one: 100x - x = 15.151515... - 0.151515....Step 4: The repeating decimal parts cancel out, giving 99x = 15.Step 5: Solve for x: x = 15/99.Step 6: Simplify 15/99 by dividing numerator and denominator by 3 to get 5/33.Verification / Alternative check:To check, convert 5/33 back to decimal form. 1/33 is approximately 0.030303..., so 5/33 is 5 times that, about 0.151515..., which matches the given decimal.Why Other Options Are Wrong:17/33 is approximately 0.515151..., which is too large. 9/11 equals about 0.818181..., with an 81 repeating pattern. 7/33 is about 0.212121..., not 0.151515.... None of these match the required repeating decimal.Common Pitfalls:Errors often occur in choosing the correct power of 10 to multiply by, or in simplifying the resulting fraction. Some candidates may incorrectly use 10x instead of 100x, which fails to align the repeating blocks properly.Final Answer:The repeating decimal 0.151515... is equal to the fraction 5/33.

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

Introduction / Context:This question is essentially the same algebraic relationship as a previous one, again using a fraction and three times its reciprocal. It checks whether you can consistently set up and solve this type of equation.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.Its reciprocal is 1/x.The condition is x + 3 * (1/x) = 31/6.We must select x from the given options.Concept / Approach:The approach is to form a quadratic equation by clearing the denominator. Solving the quadratic yields two algebraic solutions, and then we check which solution appears among the options.Step-by-Step Solution:Step 1: Write the equation: x + 3/x = 31/6.Step 2: Multiply both sides by 6x: 6x * x + 6x * (3/x) = 6x * (31/6).Step 3: Simplify to get 6x^2 + 18 = 31x.Step 4: Rearrange as 6x^2 - 31x + 18 = 0.Step 5: Compute discriminant: D = (-31)^2 - 4 * 6 * 18 = 961 - 432 = 529, whose square root is 23.Step 6: Solve for x: x = [31 ± 23] / (2 * 6) = (31 ± 23) / 12, giving x = 54/12 = 9/2 or x = 8/12 = 2/3.Verification / Alternative check:Check x = 9/2: 9/2 + 3 * (2/9) = 9/2 + 6/9 = 9/2 + 2/3 = 27/6 + 4/6 = 31/6, so 9/2 satisfies the equation. The other root x = 2/3 also satisfies it, but only 9/2 appears as an answer choice.Why Other Options Are Wrong:Options 2/9, 5/4 and 4/5 do not satisfy the equation when substituted for x. Their sums x + 3/x differ from 31/6, so they are incorrect.Common Pitfalls:Common errors include arithmetic mistakes when computing the discriminant or mishandling the quadratic formula. Some students forget to check which root is actually present among the options before deciding the final answer.Final Answer:The fraction that satisfies the condition and matches the choices is 9/2.

Question 1

The sum of a non-zero number and four times its reciprocal is 17/2. What is that number?

Accepted Answer

Introduction / Context:This problem involves a number and its reciprocal. Questions like this test algebraic manipulation skills and understanding of how a number interacts with its reciprocal, which is common in equations and ratio problems.Given Data / Assumptions:Let the non-zero number be x.The reciprocal of x is 1/x.We are told that x + 4 * (1/x) = 17/2.We must find the value of x from the given options.Concept / Approach:We translate the given verbal condition into an algebraic equation, then clear the denominator and solve the resulting quadratic equation. Finally, we check which solution matches the options.Step-by-Step Solution:Step 1: Form the equation: x + 4/x = 17/2.Step 2: Multiply both sides by 2x to eliminate the fraction: 2x * x + 2x * (4/x) = 2x * (17/2).Step 3: Simplify to get 2x^2 + 8 = 17x.Step 4: Rearrange into standard quadratic form: 2x^2 - 17x + 8 = 0.Step 5: Solve the quadratic. The discriminant is 17^2 - 4 * 2 * 8 = 289 - 64 = 225. The square root is 15. So x = (17 ± 15) / (2 * 2) = (17 ± 15) / 4.Step 6: This gives two solutions: x = 32/4 = 8 or x = 2/4 = 1/2.Verification / Alternative check:Check x = 8: 8 + 4 * (1/8) = 8 + 1/2 = 8.5 = 17/2, which satisfies the equation. Check x = 1/2: 1/2 + 4 * 2 = 1/2 + 8 = 8.5 = 17/2, which also satisfies the equation. Both numbers are mathematically valid, but only 8 appears among the given options.Why Other Options Are Wrong:Substituting x = 12, 16 or 4 into x + 4/x does not yield 17/2. For example, with x = 4 we get 4 + 4 * (1/4) = 5, not 17/2. Therefore these choices do not satisfy the condition.Common Pitfalls:A typical mistake is forgetting to multiply every term by the common denominator 2x, or solving the quadratic incorrectly. Another pitfall is assuming only one solution must exist, without checking which one matches the multiple choice options.Final Answer:The value of the number that satisfies the condition and matches the options is 8.

Question 2

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

Accepted Answer

Introduction / Context:This question uses a relationship between a fraction and its reciprocal. Such problems are common in algebra sections of aptitude tests, because they require forming and solving an equation.Given Data / Assumptions:Let the fraction be x.Its reciprocal is 1/x, with x not equal to 0.We are told that x is greater than 1/x by 9/20, so x - 1/x = 9/20.We need to find x from the given options.Concept / Approach:Translate the statement into an algebraic equation, clear the denominator and solve the resulting quadratic equation. Then match the solutions with the choices and consider which value is appropriate as a fraction in this context.Step-by-Step Solution:Step 1: Set up the equation: x - 1/x = 9/20.Step 2: Multiply both sides by 20x to clear denominators: 20x * x - 20x * (1/x) = 9 * 20x / 20.Step 3: Simplify to get 20x^2 - 20 = 9x.Step 4: Rearrange into quadratic form: 20x^2 - 9x - 20 = 0.Step 5: Compute the discriminant: D = (-9)^2 - 4 * 20 * (-20) = 81 + 1600 = 1681. The square root of 1681 is 41.Step 6: Solve for x: x = [9 ± 41] / (2 * 20) = (9 ± 41) / 40. So x = 50/40 = 5/4 or x = -32/40 = -4/5.Verification / Alternative check:Check x = 5/4: 5/4 - 4/5 = (25 - 16) / 20 = 9/20, which satisfies the equation. Check x = -4/5: -4/5 - (-5/4) also equals 9/20, but this gives a negative fraction while the context and options suggest a positive fraction.Why Other Options Are Wrong:4/5 and 3/4 are less than 1 and therefore cannot be greater than their reciprocals. 4/3 is greater than 1, but substituting x = 4/3 does not give a difference of 9/20. Only 5/4 satisfies the given relationship.Common Pitfalls:Many students forget that the equation may produce two solutions, one positive and one negative. They may also make algebraic mistakes when multiplying by the common denominator or in computing the discriminant. Always check candidate solutions in the original equation.Final Answer:The required fraction is 5/4.

Question 3

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

Accepted Answer

Introduction / Context:This question again uses a relationship between a fraction and its reciprocal, but this time the fraction is greater than twice its reciprocal. It is a typical algebraic aptitude question that requires setting up and solving a quadratic equation.Given Data / Assumptions:Let the fraction be x (non-zero).Its reciprocal is 1/x.The statement says x - 2 * (1/x) = 7/15.We must determine the correct value of x from the options.Concept / Approach:We convert the verbal statement into an algebraic equation and clear denominators. The result is a quadratic equation. Solving the quadratic and matching with the choices yields the correct fraction.Step-by-Step Solution:Step 1: Write the equation: x - 2/x = 7/15.Step 2: Multiply both sides by 15x to eliminate fractions: 15x * x - 15x * (2/x) = 7 * 15x / 15.Step 3: Simplify to get 15x^2 - 30 = 7x.Step 4: Rearrange into standard quadratic form: 15x^2 - 7x - 30 = 0.Step 5: Compute the discriminant: D = (-7)^2 - 4 * 15 * (-30) = 49 + 1800 = 1849. The square root of 1849 is 43.Step 6: Solve for x: x = [7 ± 43] / (2 * 15) = (7 ± 43) / 30, giving x = 50/30 = 5/3 or x = -36/30 = -6/5.Verification / Alternative check:Check x = 5/3: 5/3 - 2 * (3/5) = 5/3 - 6/5 = (25 - 18)/15 = 7/15, so 5/3 satisfies the original condition. The other root x = -6/5 also satisfies the algebraic equation, but it is negative and not listed among the choices. The options suggest a positive fraction, so we select 5/3.Why Other Options Are Wrong:Substituting 3/5, 3/4 or 4/3 into x - 2/x does not yield 7/15. For example, 3/5 - 2 * (5/3) is negative and far from 7/15. Therefore these values cannot be correct.Common Pitfalls:Errors often occur when multiplying by the common denominator or simplifying the quadratic. Some candidates forget to test the roots back in the original equation. Others ignore the sign of the fraction and choose a negative value even when the context points to a positive answer.Final Answer:The fraction satisfying the given condition is 5/3.

Question 4

Convert the decimal 0.0068 into a vulgar (common) fraction in lowest terms.

Accepted Answer

Introduction / Context:This question checks your ability to convert a finite decimal into a common fraction and then reduce that fraction to lowest terms. This is a basic but important skill in number systems and arithmetic.Given Data / Assumptions:Decimal: 0.0068.We must convert this decimal into a fraction.The fraction should be in simplest form.Concept / Approach:To convert a decimal to a fraction, we count the number of decimal places and use a power of 10 as the denominator. Then we simplify the fraction by dividing numerator and denominator by their greatest common divisor.Step-by-Step Solution:Step 1: Observe that 0.0068 has four digits after the decimal point.Step 2: Write 0.0068 as 68/10000 because moving the decimal four places to the right gives 68.Step 3: Now simplify 68/10000 by finding a common factor of 68 and 10000.Step 4: Notice that both are divisible by 4. 68 ÷ 4 = 17 and 10000 ÷ 4 = 2500. So 68/10000 simplifies to 17/2500.Step 5: Check if 17/2500 can be simplified further. The numerator 17 is prime and does not divide 2500, so this is already in lowest terms.Verification / Alternative check:Convert 17/2500 back to a decimal to confirm: 17 ÷ 2500 = 0.0068. This matches the original decimal, confirming that 17/2500 is correct.Why Other Options Are Wrong:7/1250 equals 0.0056, not 0.0068. 17/5000 equals 0.0034, which is too small. 7/1750 also simplifies to 1/250 and is 0.004, which does not match the given decimal. None of these alternatives produce 0.0068.Common Pitfalls:A common mistake is miscounting the number of decimal places and using 1000 instead of 10000, which leads to an incorrect fraction. Another error is forgetting to simplify the fraction, leaving it as 68/10000 instead of reducing it to lowest terms.Final Answer:The decimal 0.0068, written as a vulgar fraction in simplest form, is 17/2500.

Question 5

Which of the following fractions must be added to 5/8 to obtain a total of 1?

Accepted Answer

Introduction / Context:This question tests simple fraction addition and understanding of equivalent fractions. You are asked to find which fraction, when added to 5/8, gives a total of 1.Given Data / Assumptions:Known fraction: 5/8.Target sum: 1.Options: 6/16, 5/2, 6/24 and 6/3.Concept / Approach:If a/b + c/b = 1, then c/b = 1 - a/b. Here we first find the difference 1 - 5/8, then check which option is an equivalent fraction to that difference. Alternatively, we can convert each option to an equivalent fraction with denominator 8 and test directly.Step-by-Step Solution:Step 1: Compute 1 - 5/8. We can write 1 as 8/8, so 1 - 5/8 = 8/8 - 5/8 = 3/8.Step 2: We need a fraction equivalent to 3/8 among the choices.Step 3: Check option 6/16. Simplify 6/16 by dividing numerator and denominator by 2 to get 3/8.Step 4: Therefore, 6/16 is exactly equal to 3/8, which is what we must add to 5/8 to reach 1.Step 5: Quick check: 5/8 + 3/8 = 8/8 = 1, so the choice 6/16, which equals 3/8, works.Verification / Alternative check:We can convert each option to decimal form. 5/8 is 0.625. We need 0.375 more. 6/16 simplifies to 0.375, whereas 5/2 is 2.5, 6/24 equals 0.25 and 6/3 equals 2. None of these other options add to 0.375.Why Other Options Are Wrong:5/2 and 6/3 are both greater than 1, so adding them to 5/8 would produce a number larger than 1. 6/24 simplifies to 1/4, which is 0.25, and 0.625 + 0.25 = 0.875, not 1.Common Pitfalls:Some students do not simplify fractions and may incorrectly think 6/16 and 6/24 have the same value. Others may try to add without bringing fractions to a common denominator, leading to incorrect sums.Final Answer:The fraction that must be added to 5/8 to make 1 is 6/16, which is equivalent to 3/8.

Question 6

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

Accepted Answer

Introduction / Context:This problem involves a fraction and its reciprocal in an algebraic relationship. Such questions are popular in aptitude tests as they require forming and solving quadratic equations based on reciprocal relationships.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.Its reciprocal is 1/x.We are told that x + 3 * (1/x) = 31/6.We must find x from the given options.Concept / Approach:We convert the word statement into an equation, clear denominators, and solve the resulting quadratic equation. Then we match the solutions with the given options and select the appropriate fraction.Step-by-Step Solution:Step 1: Form the equation: x + 3/x = 31/6.Step 2: Multiply both sides by 6x to eliminate fractions: 6x * x + 6x * (3/x) = 6x * (31/6).Step 3: Simplify to get 6x^2 + 18 = 31x.Step 4: Rearrange into quadratic form: 6x^2 - 31x + 18 = 0.Step 5: Compute the discriminant: D = (-31)^2 - 4 * 6 * 18 = 961 - 432 = 529. The square root of 529 is 23.Step 6: Solve for x: x = [31 ± 23] / (2 * 6) = (31 ± 23) / 12. Thus x = 54/12 = 9/2 or x = 8/12 = 2/3.Verification / Alternative check:Check x = 9/2: 9/2 + 3 * (2/9) = 9/2 + 6/9 = 9/2 + 2/3 = 27/6 + 4/6 = 31/6. This satisfies the equation. The other root x = 2/3 also satisfies the relation, but 2/3 is not among the options, whereas 9/2 is. Therefore 9/2 is the correct answer for this multiple choice question.Why Other Options Are Wrong:Substituting 2/9, 9/4 or 4/9 into x + 3/x does not yield 31/6. They produce different sums, so those options cannot be correct.Common Pitfalls:Common mistakes include incorrect multiplication when clearing denominators, miscomputing the discriminant of the quadratic, or assuming there will be only one algebraic solution. Always check which allowed root appears in the answer choices.Final Answer:The fraction that satisfies the given condition and matches the options is 9/2.

Question 7

Arrange the fractions 3/4, 5/12, 13/16, 16/29 and 3/8 in ascending order of magnitude.

Accepted Answer

Introduction / Context:This question is another exercise in ordering fractions. Here you must compare five different fractions, some of which have fairly different denominators, and then arrange them from smallest to largest.Given Data / Assumptions:Fractions: 3/4, 5/12, 13/16, 16/29 and 3/8.We need the ascending (increasing) order.All fractions are positive.Concept / Approach:There are two practical methods: converting to decimal approximations or using a common denominator. Because one denominator is 29, decimals give a faster comparison here, while still being accurate enough.Step-by-Step Solution:Step 1: Compute 3/8 = 0.375.Step 2: Compute 5/12 ≈ 0.4167.Step 3: Compute 16/29. Using division, 16 ÷ 29 ≈ 0.5517.Step 4: Compute 3/4 = 0.75.Step 5: Compute 13/16 = 0.8125.Step 6: Arrange the decimal values: 0.375 Verification / Alternative check:Although using a common denominator is possible, it would require the least common multiple of 4, 12, 16, 29 and 8, which is large. Using decimal approximations is acceptable as long as we are careful to keep enough precision.Why Other Options Are Wrong:Option a contains 16/19 instead of 16/29, which does not match the original list. Options c and d place 13/16 before 3/4, but 0.8125 is larger than 0.75, so this is incorrect. Only option b correctly preserves the ascending order based on actual values.Common Pitfalls:Candidates sometimes misread 16/29 as 16/19 or approximate decimals too roughly, which can change the ordering of close fractions. Staying organized and writing decimals with at least four decimal places avoids such errors.Final Answer:The correct ascending order of the given fractions is 3/8 < 5/12 < 16/29 < 3/4 < 13/16.

Question 8

When the repeating decimal 0.090909... is written as a fraction in simplest form, what is that fraction?

Accepted Answer

Introduction / Context:This question tests your understanding of how to convert a repeating decimal into an exact fraction. Repeating decimals often appear in algebra and number system problems, and knowing how to handle them shows deeper conceptual understanding.Given Data / Assumptions:Repeating decimal: 0.090909..., where 09 repeats indefinitely.We must express this as a fraction in simplest form.The dots indicate that the pattern 09 continues without end.Concept / Approach:To convert a repeating decimal into a fraction, we use an algebraic method. We denote the decimal by a variable, multiply by a suitable power of 10 so that the repeating part lines up, and then subtract to eliminate the repetition.Step-by-Step Solution:Step 1: Let x = 0.090909..., with 09 repeating.Step 2: Since the repeating block has two digits, multiply both sides by 100: 100x = 9.090909...Step 3: Now subtract the original x from this new equation: 100x - x = 9.090909... - 0.090909....Step 4: On the right-hand side, the repeating parts cancel, leaving 9.Step 5: So 99x = 9, which implies x = 9/99.Step 6: Simplify 9/99 by dividing numerator and denominator by 9 to get 1/11.Verification / Alternative check:We know that 1/11 is 0.090909..., so converting back to a decimal immediately confirms that our fraction form is correct.Why Other Options Are Wrong:1/33 equals approximately 0.030303..., which has a different repeating block. 2/33 equals about 0.060606..., also incorrect. 6/11 equals about 0.545454..., which is much larger and contains 54 as the repeating pair, not 09.Common Pitfalls:A common error is misidentifying the length of the repeating block or multiplying by 10 instead of 100 in this case. Another mistake is simplifying 9/99 incorrectly or not simplifying it at all, forgetting to reduce the fraction to lowest terms.Final Answer:The repeating decimal 0.090909... is equal to the fraction 1/11.

Question 9

The sum of a fraction and ten times its reciprocal is 37/4. What is the fraction?

Accepted Answer

Introduction / Context:This problem again uses a relationship between a fraction and its reciprocal, with a larger coefficient on the reciprocal term. It is an algebraic question that reinforces skills in forming and solving quadratic equations.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.The reciprocal is 1/x.The condition is x + 10 * (1/x) = 37/4.We need to find x among the given options.Concept / Approach:We translate the condition into an equation, clear denominators, and solve the quadratic equation that results. Finally, we see which algebraic solution appears in the answer choices.Step-by-Step Solution:Step 1: Write the equation: x + 10/x = 37/4.Step 2: Multiply both sides by 4x to clear fractions: 4x * x + 4x * (10/x) = 4x * (37/4).Step 3: Simplify to get 4x^2 + 40 = 37x.Step 4: Rearrange into standard form: 4x^2 - 37x + 40 = 0.Step 5: Compute the discriminant: D = (-37)^2 - 4 * 4 * 40 = 1369 - 640 = 729. The square root of 729 is 27.Step 6: Solve for x: x = [37 ± 27] / (2 * 4) = (37 ± 27) / 8. Thus x = 64/8 = 8 or x = 10/8 = 5/4.Verification / Alternative check:Check x = 5/4: 5/4 + 10 * (4/5) = 5/4 + 8 = 5/4 + 32/4 = 37/4, which matches the given sum. The value x = 8 also satisfies the equation, because 8 + 10 * (1/8) = 8 + 1.25 = 9.25 = 37/4. However, 8 is not listed among the options, whereas 5/4 is.Why Other Options Are Wrong:Substituting 4/5, 3/4 or 4/3 into x + 10/x does not produce 37/4. Their resulting sums are quite different, so these options do not satisfy the condition.Common Pitfalls:Mistakes can occur when clearing denominators and simplifying the quadratic. Some candidates may forget that there can be two valid algebraic solutions but only one matching the provided options. Always test candidate values back in the original equation.Final Answer:The fraction that fits the condition and appears in the choices is 5/4.

Question 10

When the repeating decimal 0.151515... is written as a fraction in lowest terms, what is that fraction?

Accepted Answer

Introduction / Context:This question focuses on converting a repeating decimal with a two-digit repeating block into a fraction. Understanding this method strengthens algebraic manipulation skills and number sense.Given Data / Assumptions:Repeating decimal: 0.151515..., with 15 repeating indefinitely.We must express this decimal as a fraction in simplest form.The repeating pattern is constant throughout the decimal part.Concept / Approach:The standard technique is to assign the decimal to a variable, multiply by a power of 10 equal to the length of the repeating block, and then subtract to eliminate the repeating part. The remaining equation is easy to convert into a fraction.Step-by-Step Solution:Step 1: Let x = 0.151515..., where 15 repeats.Step 2: Since the repeating block has two digits, multiply both sides by 100: 100x = 15.151515...Step 3: Subtract the original equation from this new one: 100x - x = 15.151515... - 0.151515....Step 4: The repeating decimal parts cancel out, giving 99x = 15.Step 5: Solve for x: x = 15/99.Step 6: Simplify 15/99 by dividing numerator and denominator by 3 to get 5/33.Verification / Alternative check:To check, convert 5/33 back to decimal form. 1/33 is approximately 0.030303..., so 5/33 is 5 times that, about 0.151515..., which matches the given decimal.Why Other Options Are Wrong:17/33 is approximately 0.515151..., which is too large. 9/11 equals about 0.818181..., with an 81 repeating pattern. 7/33 is about 0.212121..., not 0.151515.... None of these match the required repeating decimal.Common Pitfalls:Errors often occur in choosing the correct power of 10 to multiply by, or in simplifying the resulting fraction. Some candidates may incorrectly use 10x instead of 100x, which fails to align the repeating blocks properly.Final Answer:The repeating decimal 0.151515... is equal to the fraction 5/33.

Question 11

Reduce the fraction 2714/5074 to lowest terms.

Accepted Answer

Introduction / Context:This question checks whether you can simplify a relatively large fraction to its lowest terms. Reducing fractions is often needed in data interpretation, ratio problems and simplifying algebraic expressions.Given Data / Assumptions:Fraction: 2714/5074.We need to express this fraction in lowest terms.We assume both numerator and denominator are positive integers.Concept / Approach:To reduce a fraction to lowest terms, we divide both numerator and denominator by their greatest common divisor. The main task is to find that greatest common divisor, often by systematic trial with prime factors or by the Euclidean algorithm.Step-by-Step Solution:Step 1: Look for common small prime factors. Both 2714 and 5074 are even, so they are at least divisible by 2.Step 2: Divide numerator and denominator by 2: 2714 ÷ 2 = 1357 and 5074 ÷ 2 = 2537.Step 3: Now the fraction becomes 1357/2537.Step 4: Check if 1357 and 2537 have any further common factors. It turns out they are both divisible by 59 or a common factor that simplifies to give a familiar ratio.Step 5: Dividing properly, 2714/5074 simplifies exactly to 23/43. This can be confirmed by cross multiplication: 2714 * 43 = 116, and 5074 * 23 = 116, which matches, confirming equivalence.Verification / Alternative check:You can also verify by direct division: 2714 ÷ 23 = 118 and 5074 ÷ 43 = 118, so both numerator and denominator share the factor 118. Dividing by 118 yields 23/43 directly.Why Other Options Are Wrong:17/23, 29/43 and 31/37 do not equal 2714/5074 when cross multiplied. For example, comparing 2714/5074 with 23/43 works, but cross multiplying with 17/23 gives mismatched products, so those fractions are not equivalent.Common Pitfalls:Many candidates stop after dividing by 2 and incorrectly assume that 1357/2537 is already in lowest terms. Others may guess simplifications instead of systematically checking for common factors. Using a structured factorization or Euclidean algorithm is more reliable.Final Answer:The fraction 2714/5074 in lowest terms is 23/43.

Question 12

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

Accepted Answer

Introduction / Context:This question is essentially the same algebraic relationship as a previous one, again using a fraction and three times its reciprocal. It checks whether you can consistently set up and solve this type of equation.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.Its reciprocal is 1/x.The condition is x + 3 * (1/x) = 31/6.We must select x from the given options.Concept / Approach:The approach is to form a quadratic equation by clearing the denominator. Solving the quadratic yields two algebraic solutions, and then we check which solution appears among the options.Step-by-Step Solution:Step 1: Write the equation: x + 3/x = 31/6.Step 2: Multiply both sides by 6x: 6x * x + 6x * (3/x) = 6x * (31/6).Step 3: Simplify to get 6x^2 + 18 = 31x.Step 4: Rearrange as 6x^2 - 31x + 18 = 0.Step 5: Compute discriminant: D = (-31)^2 - 4 * 6 * 18 = 961 - 432 = 529, whose square root is 23.Step 6: Solve for x: x = [31 ± 23] / (2 * 6) = (31 ± 23) / 12, giving x = 54/12 = 9/2 or x = 8/12 = 2/3.Verification / Alternative check:Check x = 9/2: 9/2 + 3 * (2/9) = 9/2 + 6/9 = 9/2 + 2/3 = 27/6 + 4/6 = 31/6, so 9/2 satisfies the equation. The other root x = 2/3 also satisfies it, but only 9/2 appears as an answer choice.Why Other Options Are Wrong:Options 2/9, 5/4 and 4/5 do not satisfy the equation when substituted for x. Their sums x + 3/x differ from 31/6, so they are incorrect.Common Pitfalls:Common errors include arithmetic mistakes when computing the discriminant or mishandling the quadratic formula. Some students forget to check which root is actually present among the options before deciding the final answer.Final Answer:The fraction that satisfies the condition and matches the choices is 9/2.

Question 13

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

Accepted Answer

Introduction / Context:This algebraic question generalizes the earlier patterns by using twice a fraction and three times its reciprocal. It is designed to test your comfort with forming equations from verbal statements and solving quadratics.Given Data / Assumptions:Let the fraction be x, with x not equal to 0.The reciprocal is 1/x.The condition given is 2x + 3 * (1/x) = 29/3.We must determine the value of x from the answer choices.Concept / Approach:We transform the condition into a quadratic equation by clearing denominators. After solving the quadratic, we test the resulting values against the options to identify the correct fraction.Step-by-Step Solution:Step 1: Write the given relationship: 2x + 3/x = 29/3.Step 2: Multiply through by 3x to clear denominators: 3x * 2x + 3x * (3/x) = 3x * (29/3).Step 3: Simplify to obtain 6x^2 + 9 = 29x.Step 4: Rearrange into standard quadratic form: 6x^2 - 29x + 9 = 0.Step 5: Compute the discriminant: D = (-29)^2 - 4 * 6 * 9 = 841 - 216 = 625. The square root of 625 is 25.Step 6: Use the quadratic formula: x = [29 ± 25] / (2 * 6) = (29 ± 25) / 12. This gives x = 54/12 = 9/2 or x = 4/12 = 1/3.Verification / Alternative check:Check x = 9/2: 2 * (9/2) + 3 * (2/9) = 9 + 6/9 = 9 + 2/3 = 27/3 + 2/3 = 29/3. This matches the given condition. The other root x = 1/3 yields 2 * (1/3) + 3 * 3 = 2/3 + 9 = 29/3 as well, but 1/3 is not listed among the options.Why Other Options Are Wrong:Substituting 2/9, 5/4 or 4/5 into 2x + 3/x does not produce 29/3. Their resulting values are different, so they do not satisfy the equation.Common Pitfalls:Errors can occur when multiplying by the common denominator, especially if a term is accidentally omitted. Another pitfall is incorrect calculation of the discriminant, which leads to wrong roots. Finally, some students forget to check which algebraic solution appears in the multiple choice list.Final Answer:The fraction that satisfies the relationship and appears in the options is 9/2.

Question 14

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

Accepted Answer

Introduction / Context:This aptitude question on fractions and algebra tests your ability to translate a word statement into an equation involving a number and its reciprocal. Such problems are very common in bank exams, SSC, and other competitive tests, and they check whether you can set up and solve a quadratic equation correctly.Given Data / Assumptions:Let the fraction be x, where x is a non-zero rational number.Four times the fraction is written as 4x.Six times the reciprocal of the fraction is written as 6 * (1/x).The sum of these two expressions is given to be 11.Concept / Approach:The key idea is to convert the English statement into an algebraic equation. Once we have the equation 4x + 6/x = 11, we clear the denominator and obtain a quadratic equation in x. Solving the quadratic gives two possible values, and then we choose the one that matches the idea of a fraction less than 1 (since typical exam phrasing suggests so) and is present in the options.Step-by-Step Solution:Step 1: Let the required fraction be x.Step 2: Four times the fraction is 4x.Step 3: Six times its reciprocal is 6 * (1/x).Step 4: According to the question, 4x + 6 * (1/x) = 11.Step 5: Multiply through by x (x ≠ 0) to clear the denominator: 4x^2 + 6 = 11x.Step 6: Rearrange to standard quadratic form: 4x^2 - 11x + 6 = 0.Step 7: Factorize: 4x^2 - 11x + 6 = (4x - 3)(x - 2) = 0.Step 8: Therefore, x = 3/4 or x = 2.Step 9: Out of these, 3/4 is a proper fraction and appears in the options, so the required fraction is 3/4.Verification / Alternative check:Check x = 3/4: Four times the fraction = 4 * 3/4 = 3.Reciprocal of 3/4 is 4/3, so six times reciprocal = 6 * 4/3 = 8.Sum = 3 + 8 = 11, which matches the given condition. Hence 3/4 is correct. The value 2 also satisfies the equation but is not offered as a valid option and does not fit the natural reading of a fraction in this context.Why Other Options Are Wrong:Option b (4/3) is greater than 1. Substituting 4/3 will not give sum 11. Option c (4/7) and option d (7/4) similarly fail to satisfy the equation 4x + 6/x = 11 when substituted, so they are incorrect distractors.Common Pitfalls:Candidates often make errors by forgetting to multiply every term by x when clearing the denominator, which leads to an incorrect quadratic equation. Another mistake is to assume only one root will arise and to ignore that quadratics usually have two solutions; however, you must then check which solution suits the context and appears in the options. Mis-simplifying fractions such as 6 * 4/3 is another frequent source of small calculation errors.Final Answer:The required fraction is 3/4.

Question 15

Which part contains the fractions in ascending order?

Accepted Answer

Concept & Strategy: To arrange fractions in ascending order (smallest to largest), either convert them into decimal values or analyze the numerical gap between their numerators and denominators. Step-by-Step Solution: Let's evaluate the approximate decimal value for each unique fraction: • (11/14) \approx 0.785 • (16/19) \approx 0.842 • (19/21) \approx 0.904 Comparing the resulting decimal values: 0.785 (11/14). When comparing fractions with similar numbers, a smaller gap (like 2 for (19/21)) generally indicates a fraction closer to 1. Thus, (19/21) is the largest. The correct ascending sequence is (11/14), (16/19), (19/21). Common Pitfall: Miscalculating decimal values during long division, especially with prime denominators like 19. Using the gap analysis shortcut saves time and reduces calculation errors. Final Answer: Therefore, the correct answer is option (a).

Question 16

Which of the following fractions is greater than $$\frac{3}{4}$$ and less than $$\frac{5}{6}$$?

Accepted Answer

Concept & Logic: To find a fraction that falls within a specific range, convert the lower and upper bounds into percentages or decimals, then evaluate the given options against those bounds. Step-by-Step Solution: • First, establish the target range by converting the boundary fractions to decimals: • Lower bound: (3/4) = 0.75 • Upper bound: (5/6) \approx 0.833 • We need an option whose value is strictly between 0.75 and 0.833. • Next, convert the given options into decimals: (a) (1/2) = 0.50 (b) (2/3) \approx 0.667 (c) (4/5) = 0.80 (d) (9/10) = 0.90 • Comparing the options to our range (0.75 < x < 0.833): • 0.50 is too small. • 0.667 is too small. • 0.80 falls perfectly within the range. • 0.90 is too large. Exam Strategy & Shortcut: Percentage Equivalents: Memorizing standard fraction-to-percentage conversions makes this a 5-second question. Bounds: 75% and 83.33%. Options: 50%, 66.66%, 80%, 90%. It is immediately obvious that only 80% ((4/5)) fits in the required gap. Common Pitfall: Attempting to equalize the denominators for all six fractions (the two bounds plus the four options) by finding a massive common multiple. This is entirely unnecessary and wastes critical time. Final Answer: Therefore, the correct answer is option (c).

Question 17

The arrangement of rational numbers $$\frac{-7}{10}, \frac{5}{-8}, \frac{2}{-3}$$ in ascending order is

Accepted Answer

Concept & Strategy: To arrange negative fractions in ascending order, standardize the negative signs to the numerator, calculate their decimal values (or make denominators equal), and order them from the largest magnitude negative number to the smallest. Step-by-Step Solution: • First, rewrite the fractions in a standard format: (-7/10), (-5/8), (-2/3) • Next, convert them into approximate decimal values for easy comparison: • (-7/10) = -0.700 • (-5/8) = -0.625 • (-2/3) \approx -0.666 • Remember that for negative numbers, a larger absolute value means a smaller actual value. • Comparing the decimals: -0.700 < -0.666 < -0.625 • Replacing them with their original fractional forms: (-7/10) < (2/-3) < (5/-8) Exam Strategy & Shortcut: Common Denominator (LCM): If you dislike decimals, find the LCM of 10, 8, and 3, which is 120. • (-7 × 12/120) = (-84/120) • (-5 × 15/120) = (-75/120) • (-2 × 40/120) = (-80/120) Comparing numerators directly: -84 < -80 < -75. The correct order instantly reveals itself as (-7/10), (2/-3), (5/-8). Common Pitfall: Treating negative fractions like positive ones. Students often order them as 0.625, 0.666, 0.700 and accidentally arrange them in descending order instead of ascending. Always double-check the negative sign rules. Final Answer: Therefore, the correct answer is option (d).

Question 18

$0.3 + 3 + 3.33 + 3.3 + 3.03 + 333 = $ $x$

Accepted Answer

Concept & Strategy: This is a straightforward addition problem evaluating your ability to align decimal points correctly. The core logic is to align numbers by their place values before summing. Step-by-Step Solution: • Write the numbers vertically, strictly aligning the decimal points. Pad with zeros to avoid confusion: • 000.30 • 003.00 • 003.33 • 003.30 • 003.03 • 333.00 • Sum the hundredths column: • 0 + 0 + 3 + 0 + 3 + 0 = 6 • Sum the tenths column: • 3 + 0 + 3 + 3 + 0 + 0 = 9 • Sum the units column: • 0 + 3 + 3 + 3 + 3 + 3 = 15 (Write 5, carry over 1) • Sum the tens column: • 0 + 0 + 0 + 0 + 0 + 3 + 1 (carry) = 4 • Sum the hundreds column: • 3 • Putting it all together, the final sum is 345.96. Exam Strategy & Shortcut: Unit Digit & Quick Estimation: Instead of a full calculation, split the integers and decimals. Integer sum: 3 + 3 + 3 + 3 + 333 = 345. Decimal sum: 0.30 + 0.33 + 0.30 + 0.03 = 0.96. Total: 345.96. Looking at the options, (a) is 345.99 and (b) is 355.96. Neither matches 345.96 perfectly. Hence, choose "None of these". Common Pitfall: The most common mistake is misaligning the decimal points (e.g., treating 3 as 0.3 or 333 as 3.33) or making basic addition errors in the final tens/units place which leads to picking trap option (b) 355.96. Final Answer: Therefore, the correct answer is None of these.

Question 19

$636.66 + 366.36 + 363.33 = $ $x$

Accepted Answer

Concept & Strategy: This question tests precision in decimal addition. The fundamental rule is to ensure all decimal points are aligned vertically before adding the respective place values. Step-by-Step Solution: • Align the decimals vertically for accurate calculation: • 636.66 • 366.36 • 363.33 • Add the hundredths column: • 6 + 6 + 3 = 15 (Write 5, carry over 1) • Add the tenths column: • 6 + 3 + 3 + 1 (carry) = 13 (Write 3, carry over 1) • Place the decimal point. • Add the units column: • 6 + 6 + 3 + 1 (carry) = 16 (Write 6, carry over 1) • Add the tens column: • 3 + 6 + 6 + 1 (carry) = 16 (Write 6, carry over 1) • Add the hundreds column: • 6 + 3 + 3 + 1 (carry) = 13 • The exact total is 1366.35. Exam Strategy & Shortcut: Unit Digit Verification: Calculate only the last two decimal digits to save time. Hundredths: 6 + 6 + 3 = 15 (ends in 5, carry 1). Tenths: 6 + 3 + 3 + 1 = 13 (ends in 35). The answer must end in .35. Looking at the options, only (a) 1336.35 ends in .35. Now do a quick hundreds estimation: 600 + 300 + 300 = 1200, plus the tens 36+66+63 \approx 160, so the total is around 1360. Option (a) is 1336, which is too low. The true answer 1366.35 isn't there, meaning we can quickly select "None of these". Common Pitfall: Choosing a "close" answer. Students might calculate 1366.35 but select option (c) 1366.25 or option (a) 1336.35 out of haste due to visual similarity, instead of confidently picking "None of these". Final Answer: Therefore, the correct answer is None of these.

Question 20

$10.0001 + 9.9999 - 8.9995 = $ $x$

Accepted Answer

Concept & Strategy: This problem evaluates mastery over the order of operations involving decimal addition and subtraction. Performing operations left-to-right ensures proper management of carrying and borrowing. Step-by-Step Solution: • First, add the first two terms together by aligning their decimal points: • 10.0001 • + \ 9.9999 • Calculate the sum step-by-step from right to left: • 1 + 9 = 10 \rightarrow 0, carry 1 • 0 + 9 + 1 = 10 \rightarrow 0, carry 1 • 0 + 9 + 1 = 10 \rightarrow 0, carry 1 • 0 + 9 + 1 = 10 \rightarrow 0, carry 1 • 10 + 9 + 1 = 20 • Intermediate Sum = 20.0000 • Next, subtract the third term from this intermediate sum: • 20.0000 - 8.9995 • Perform standard column borrowing across the zeros: • 20.0000 - 8.9995 = 11.0005 Exam Strategy & Shortcut: Complementary Visual Grouping: Notice that 9.9999 - 8.9995 = 1.0004. Now, simply add this result back to the first term: 10.0001 + 1.0004 = 11.0005. By changing the order of calculation dynamically, you turn a multi-stage carry-and-borrow problem into a trivial mental addition step. Common Pitfall: A common error lies in borrowing incorrectly from 20.0000, resulting in values like 11.0001 or forgetting to decrement the units place during continuous place-value borrowing. Final Answer: Therefore, the correct answer is 11.0005.

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