Fractions & Decimals — Practice Set
Bank code: ARI-FRC · Section: Arithmetic · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5
Stay in fractions: cross-multiply to compare, and reduce before you judge a denominator.
Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.
Part A — Questions
Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given
Level 1 · Easy — 5 questions · ~4 min
warm-up — these must be automatic
Q1 · ARI-FRC-001 · MCQ · 50s
What is 3/4 + 1/6?
(A) 2/5
(B) 5/12
(C) 7/12
(D) 11/12
(E) 1/8
Q2 · ARI-FRC-007 · Numeric Entry · 45s
Write 5/8 as a decimal. Enter your answer as a number.
Numeric entry — write the number.
Q3 · ARI-FRC-010 · MCQ · 45s
What is 0.4 x 0.06?
(A) 0.46
(B) 2.4
(C) 0.24
(D) 0.024
(E) 0.0024
Q4 · ARI-FRC-018 · QC · 50s
n < 0
Column A: n/3 Column B: n/4
Q5 · ARI-FRC-020 · Select all that apply · 60s
Which of the following fractions are greater than 1/2? Select all that apply.
(A) 3/7
(B) 5/9
(C) 7/13
(D) 9/19
(E) 6/11
Level 2 · Medium — 8 questions · ~10 min
two or three steps, one planted trap each
Q6 · ARI-FRC-021 · MCQ · 85s
What is the value of (2/3 + 1/4) / (1/2 - 1/3)?
(A) 11/72
(B) 2/11
(C) 3/2
(D) 11/12
(E) 11/2
Q7 · ARI-FRC-026 · MCQ · 80s
A tank is 2/5 full. After 30 gallons are added, it is 4/5 full. What is the total capacity of the tank, in gallons?
(A) 12
(B) 37.5
(C) 50
(D) 75
(E) 150
Q8 · ARI-FRC-027 · QC · 70s
0 < x < 1
Column A: x + 1/x Column B: 2
Q9 · ARI-FRC-029 · MCQ · 75s
What is 0.00045 / 0.009?
(A) 0.005
(B) 0.05
(C) 0.5
(D) 5
(E) 50
Q10 · ARI-FRC-036 · QC · 80s
a and b are positive integers with a > b.
Column A: a/b Column B: (a + 1)/(b + 1)
Q11 · ARI-FRC-048 · MCQ · 75s
What is 3 1/3 x 2 1/4?
(A) 5 7/12
(B) 6 1/12
(C) 6 1/4
(D) 6 3/4
(E) 7 1/2
Q12 · ARI-FRC-049 · Numeric Entry · 75s
What fraction of 3/4 is 5/8? Enter your answer as a fraction in lowest terms.
Numeric entry — write the number.
Q13 · ARI-FRC-050 · Select all that apply · 90s
Let n/d be a fraction in which n and d are positive integers with n < d. Which of the following must be true? Select all that apply.
(A) (n/d)^2 < n/d
(B) d/n > 1
(C) (n + 1)/(d + 1) > n/d
(D) sqrt(n/d) < n/d
(E) (n/d)^3 < (n/d)^2
Level 3 · Hard — 12 questions · ~19 min
where 162+ is won or lost
Q14 · ARI-FRC-051 · MCQ · 90s
What is the value of 1 / (1 + 1 / (1 + 1/2))?
(A) 2/3
(B) 3/2
(C) 3/5
(D) 5/3
(E) 2/5
Q15 · ARI-FRC-052 · QC · 95s
0 < x < 1
Column A: x / (1 - x) Column B: x^2 / (1 - x^2)
Q16 · ARI-FRC-053 · Numeric Entry · 100s
The block 142857 repeats forever in the decimal 0.142857142857142857... Express this number as a fraction in lowest terms. Enter your answer in the form numerator/denominator.
Numeric entry — write the number.
Q17 · ARI-FRC-054 · MCQ · 100s
p, q, r and s are positive numbers with p/q = r/s. Which of the following must also equal p/q?
(A) (p + r)/(q + s)
(B) (p + q)/(r + s)
(C) (p + s)/(q + r)
(D) p^2/q^2
(E) (p + 1)/(q + 1)
Q18 · ARI-FRC-057 · Numeric Entry · 90s
a and b are positive numbers with 1/a + 1/b = 1/6 and ab = 192. What is a + b? Enter your answer as a number.
Numeric entry — write the number.
Q19 · ARI-FRC-058 · MCQ · 105s
What is the value of (0.1 + 0.2 + 0.3 + ... + 0.9)^2 - (0.1^2 + 0.2^2 + 0.3^2 + ... + 0.9^2)?
(A) 0
(B) 1.65
(C) 17.4
(D) 20.25
(E) 26.4
Q20 · ARI-FRC-062 · MCQ · 90s
What is the value of 1/(1 x 2) + 1/(2 x 3) + 1/(3 x 4) + ... + 1/(9 x 10)?
(A) 1/2
(B) 9/10
(C) 1
(D) 1/10
(E) 10/9
Q21 · ARI-FRC-064 · MCQ · 85s
Let x = 0.999... , the decimal in which the digit 9 repeats forever. Which of the following is true?
(A) x < 1
(B) x = 1
(C) x > 1
(D) x is irrational
(E) x cannot be written as a ratio of two integers
Q22 · ARI-FRC-065 · Numeric Entry · 95s
Positive numbers a, b, c, d and e satisfy a/b = 3/4, b/c = 2/5, c/d = 5/6 and d/e = 4/9. What is a/e? Enter your answer as a fraction in lowest terms.
Numeric entry — write the number.
Q23 · ARI-FRC-066 · Select all that apply · 100s
Which of the following fractions have terminating decimal expansions? Select all that apply.
(A) 3/14
(B) 7/8
(C) 18/48
(D) 11/12
(E) 21/35
Q24 · ARI-FRC-068 · MCQ · 95s
Pipe A alone fills a tank in 6 hours and pipe B alone fills the same tank in 4 hours. If both pipes run together for 1.5 hours, what fraction of the tank is filled?
(A) 1/5
(B) 3/10
(C) 5/12
(D) 5/8
(E) 5/6
Q25 · ARI-FRC-070 · Select all that apply · 105s
Which of the following conversions from a recurring decimal to a fraction are correct? Select all that apply.
(A) 0.111... = 1/9
(B) 0.121212... = 4/33
(C) 0.363636... = 9/25
(D) 0.1666... = 16/99
(E) 0.454545... = 5/11
Level 4 · Extreme — 5 questions · ~10 min
165+ — expect to need the insight, not the grind
Q26 · ARI-FRC-069 · Numeric Entry · 125s
What is the value of (1 - 1/4)(1 - 1/9)(1 - 1/16) ... (1 - 1/100), where the factors are 1 - 1/n^2 for n = 2, 3, 4, ..., 10? Enter your answer as a fraction in lowest terms.
Numeric entry — write the number.
Q27 · ARI-FRC-072 · QC · 105s
a and b are digits, not both zero, and x = 0.ababab... , the decimal in which the two-digit block ab repeats forever.
Column A: x Column B: (10a + b)/99
Q28 · ARI-FRC-077 · Numeric Entry · 115s
A sequence is defined by a(1) = 1 and a(n) = a(n-1) / (1 + a(n-1)) for every integer n > 1. What is a(20)? Enter your answer as a fraction in lowest terms.
Numeric entry — write the number.
Q29 · ARI-FRC-078 · Select all that apply · 130s
Which of the following fractions have a decimal expansion whose repeating block is exactly 6 digits long? Select all that apply.
(A) 1/7
(B) 1/11
(C) 1/13
(D) 1/14
(E) 1/9
Q30 · ARI-FRC-080 · QC · 130s
Column A: 1/(2^2 - 1) + 1/(3^2 - 1) + 1/(4^2 - 1) + ... + 1/(10^2 - 1), the sum of 1/(n^2 - 1) for n = 2, 3, 4, ..., 10 Column B: 13/20
Part B — Answer Key
| Q | ID | Level | Type | Answer |
|---|---|---|---|---|
| 1 | ARI-FRC-001 |
easy | MCQ | D |
| 2 | ARI-FRC-007 |
easy | Numeric Entry | 0.625 |
| 3 | ARI-FRC-010 |
easy | MCQ | D |
| 4 | ARI-FRC-018 |
easy | QC | B |
| 5 | ARI-FRC-020 |
easy | Select all that apply | B, C, E |
| 6 | ARI-FRC-021 |
medium | MCQ | E |
| 7 | ARI-FRC-026 |
medium | MCQ | D |
| 8 | ARI-FRC-027 |
medium | QC | A |
| 9 | ARI-FRC-029 |
medium | MCQ | B |
| 10 | ARI-FRC-036 |
medium | QC | A |
| 11 | ARI-FRC-048 |
medium | MCQ | E |
| 12 | ARI-FRC-049 |
medium | Numeric Entry | 5/6 |
| 13 | ARI-FRC-050 |
medium | Select all that apply | A, B, C, E |
| 14 | ARI-FRC-051 |
hard | MCQ | C |
| 15 | ARI-FRC-052 |
hard | QC | A |
| 16 | ARI-FRC-053 |
hard | Numeric Entry | 1/7 |
| 17 | ARI-FRC-054 |
hard | MCQ | A |
| 18 | ARI-FRC-057 |
hard | Numeric Entry | 32 |
| 19 | ARI-FRC-058 |
hard | MCQ | C |
| 20 | ARI-FRC-062 |
hard | MCQ | B |
| 21 | ARI-FRC-064 |
hard | MCQ | B |
| 22 | ARI-FRC-065 |
hard | Numeric Entry | 1/9 |
| 23 | ARI-FRC-066 |
hard | Select all that apply | B, C, E |
| 24 | ARI-FRC-068 |
hard | MCQ | D |
| 25 | ARI-FRC-070 |
hard | Select all that apply | A, B, E |
| 26 | ARI-FRC-069 |
extreme hard | Numeric Entry | 11/20 |
| 27 | ARI-FRC-072 |
extreme hard | QC | C |
| 28 | ARI-FRC-077 |
extreme hard | Numeric Entry | 1/20 |
| 29 | ARI-FRC-078 |
extreme hard | Select all that apply | A, C, D |
| 30 | ARI-FRC-080 |
extreme hard | QC | A |
Part C — Worked Solutions
Q1 · ARI-FRC-001 — Answer: D
Step 1 — find the LCD. The denominators are 4 and 6, so the LCD is 12. Step 2 — rescale BOTH parts of each fraction. 3/4 = (3 x 3)/(4 x 3) = 9/12. 1/6 = (1 x 2)/(6 x 2) = 2/12. Step 3 — add the numerators only, keeping the common denominator: 9/12 + 2/12 = 11/12. Check: 3/4 = 0.75 and 1/6 = 0.1667, so the sum is about 0.9167, and 11/12 = 0.9167. Correct. Answer: 11/12.
Trap. Adding across - (3 + 1)/(4 + 6) = 4/10 = 2/5, which is choice A. That is never how fractions add. Choice B, 5/12, comes from the half-conversion 3/12 + 2/12: the denominator was rescaled but the numerator was left alone. Choice C, 7/12, is the correctly rescaled 9/12 - 2/12, i.e. subtracting the numerators instead of adding them. Choice E, 1/8, is the product 3/4 x 1/6 rather than the sum.
Q2 · ARI-FRC-007 — Answer: 0.625
Step 1 — check that it will terminate. 8 = 2^3, only 2s, so the decimal stops. Good, no repeating block to worry about. Step 2 — force the denominator to a power of 10. 8 x 125 = 1000, so multiply top and bottom by 125: 5/8 = (5 x 125)/(8 x 125) = 625/1000 = 0.625. Step 3 — or just divide: 5.000 divided by 8 gives 0.625. Check: 0.625 x 8 = 5. Correct. Answer: 0.625.
Trap. Writing 0.58 by lining the digits up as they appear in '5/8'. A fraction bar is a division sign, not a decimal point. The other slip is rounding to 0.63 when the question asks for the exact value - 5/8 terminates exactly, so nothing needs rounding.
Q3 · ARI-FRC-010 — Answer: D
Step 1 — ignore the decimal points and multiply as integers: 4 x 6 = 24. Step 2 — count the decimal places in the two factors. 0.4 has 1 place, 0.06 has 2 places, so the answer must have 1 + 2 = 3 places. Step 3 — write 24 with three decimal places: 0.024. Check by fractions: 0.4 x 0.06 = (4/10) x (6/100) = 24/1000 = 0.024. Correct. Answer: 0.024.
Trap. Miscounting places. Choice C, 0.24, uses two places instead of three; choice B, 2.4, uses one; choice E, 0.0024, uses four. Choice A, 0.46, is the sum 0.4 + 0.06 rather than the product. Add the decimal places of the factors - do not guess from how the numbers look.
Q4 · ARI-FRC-018 — Answer: B
Step 1 — test a convenient value. Let n = -12. Then n/3 = -4 and n/4 = -3. Step 2 — compare on the number line: -3 sits to the right of -4, so -3 is greater. Column B is greater. Step 3 — confirm it always holds. n/3 - n/4 = 4n/12 - 3n/12 = n/12. Since n < 0, n/12 < 0, so Column A - Column B is always negative. Column B is greater for every negative n.
Trap. Carrying over the positive-number rule 'dividing by the smaller number gives the bigger result' and choosing A. That rule holds only for n > 0. Dividing a negative number by 3 pushes it FURTHER from zero than dividing it by 4 does, so n/3 is the more negative - and more negative means smaller. Note also that the stem pins n < 0, so D is wrong: the relationship is fully determined.
Q5 · ARI-FRC-020 — Answer: B, C, E
Cross-multiply each against 1/2. For a/b vs 1/2 with b > 0, compare 2a against b: if 2a > b the fraction is bigger than 1/2. No decimals needed. A - 3/7: 2 x 3 = 6 vs 7. 6 < 7, so 3/7 < 1/2. FALSE. B - 5/9: 2 x 5 = 10 vs 9. 10 > 9, so 5/9 > 1/2. TRUE. C - 7/13: 2 x 7 = 14 vs 13. 14 > 13, so 7/13 > 1/2. TRUE. D - 9/19: 2 x 9 = 18 vs 19. 18 < 19, so 9/19 < 1/2. FALSE. E - 6/11: 2 x 6 = 12 vs 11. 12 > 11, so 6/11 > 1/2. TRUE. Answer: B, C, E.
Trap. Eyeballing size instead of doubling the numerator. C and D look almost identical in shape, but 7/13 clears 1/2 and 9/19 misses it - the only thing that matters is whether twice the top beats the bottom. Skipping C because 7/13 'looks like about a half' is the most common miss here.
Q6 · ARI-FRC-021 — Answer: E
A complex fraction is one division. Simplify the top, simplify the bottom, then divide. Step 1 — top: LCD of 3 and 4 is 12. 2/3 = 8/12 and 1/4 = 3/12, so the top is 8/12 + 3/12 = 11/12. Step 2 — bottom: LCD of 2 and 3 is 6. 1/2 = 3/6 and 1/3 = 2/6, so the bottom is 3/6 - 2/6 = 1/6. Step 3 — divide by flipping the SECOND fraction: (11/12) / (1/6) = 11/12 x 6/1 = 66/12 = 11/2. Check in decimals: 0.9167 / 0.1667 = 5.5, and 11/2 = 5.5. Correct. Answer: 11/2.
Trap. Choice A, 11/72, is multiplying by 1/6 instead of dividing by it - the divisor must be flipped. Choice D, 11/12, is stopping after the numerator. Choice C, 3/2, comes from rescaling only the denominators in the top (2/12 + 1/12 = 3/12) and then dividing by 1/6. Choice B, 2/11, is the whole quotient turned upside down.
Q7 · ARI-FRC-026 — Answer: D
Step 1 — find what fraction of the tank the 30 gallons represents. The level moved from 2/5 to 4/5, a rise of 4/5 - 2/5 = 2/5 of the tank. Step 2 — so (2/5) x C = 30, where C is the capacity. Step 3 — divide by flipping: C = 30 / (2/5) = 30 x 5/2 = 150/2 = 75. Check: 2/5 of 75 = 30 gallons in the tank; add 30 to get 60; and 60/75 = 4/5. Correct. Answer: 75 gallons.
Trap. Choice A, 12, is 30 x 2/5 - multiplying when the flip-and-multiply step calls for 30 x 5/2. Choice B, 37.5, divides by 4/5, using the FINAL level instead of the change. Choice C, 50, divides by 3/5, the empty part rather than the part that was filled. Choice E, 150, divides by 1/5, treating the 30 gallons as one fifth of the tank instead of two fifths.
Q8 · ARI-FRC-027 — Answer: A
Step 1 — a proper fraction and its reciprocal sit on opposite sides of 1. For 0 < x < 1 the reciprocal 1/x is greater than 1: x = 1/4 gives 1/x = 4, x = 0.9 gives 1/x = 1.111... Step 2 — do not test values one at a time - subtract Column B and combine over the common denominator x: x + 1/x - 2 = (x^2 + 1 - 2x)/x = (x - 1)^2 / x. Step 3 — the numerator (x - 1)^2 is a square, and it is strictly positive because the stem excludes x = 1. The denominator x is positive. So the whole expression is positive, i.e. x + 1/x > 2 for every x in the range. Check x = 1/2: 0.5 + 2 = 2.5 > 2. Check x = 0.9: 0.9 + 1.1111 = 2.0111 > 2. Check x = 0.99: 0.99 + 1.0101 = 2.0001 > 2. The gap shrinks as x approaches 1 but never reaches zero. Column A is greater.
Trap. Choosing C by testing x = 1, where x + 1/x = 1 + 1 = 2 exactly - but the stem excludes 1, and that endpoint is the only place the columns tie. Choosing B by rounding the reciprocal down: at x = 0.9 a student who calls 1/0.9 'about 1' gets 1.9, which is less than 2; the true value is 1.1111 and the sum is 2.0111. Choosing D because x is unspecified is the third error: the sign of (x - 1)^2 / x is fixed across the whole interval, so the comparison resolves.
Q9 · ARI-FRC-029 — Answer: B
Step 1 — clear the decimal from the DIVISOR by shifting both numbers the same number of places. Moving each three places right turns 0.009 into 9 and 0.00045 into 0.45. Step 2 — the quotient is unchanged: 0.00045/0.009 = 0.45/9. Step 3 — divide: 0.45/9 = 0.05. Check by multiplying back: 0.05 x 0.009 = 0.00045 (5 x 9 = 45, and 2 + 3 = 5 decimal places). Correct. A cleaner route: 0.00045/0.009 = (45/100000)/(9/1000) = 45/100000 x 1000/9 = 45/900 = 1/20 = 0.05. Answer: 0.05.
Trap. Dividing 45 by 9 to get 5 and then guessing where the point goes. Choice D, 5, ignores the decimals entirely; choices A and C are the same digit shifted one place the wrong way; choice E, 50, comes from shifting the dividend and the divisor by different amounts. Shift BOTH numbers by the same count, then divide once.
Q10 · ARI-FRC-036 — Answer: A
Step 1 — both denominators b and b + 1 are positive, so cross-multiplication is legal and does not flip the inequality. Step 2 — cross-multiply: a(b + 1) vs b(a + 1), i.e. ab + a vs ab + b. Step 3 — cancel ab from both sides: the comparison is just a vs b. The stem says a > b, so Column A is greater. Always. Check with a = 5, b = 2: 5/2 = 2.5 and 6/3 = 2. Check with a = 100, b = 99: 100/99 = 1.0101 and 101/100 = 1.01. Column A wins both. The principle: adding the same positive amount to the top and bottom drags a fraction TOWARD 1. Here a/b > 1, so being dragged toward 1 means getting smaller.
Trap. Testing one pair of numbers, seeing the two values come out close, and choosing D. The algebra settles it in one line: after cancelling ab the whole question reduces to a vs b, which the stem has already answered. Choosing B is the mirror error - it would be correct only for a PROPER fraction (a < b), where adding 1 to both raises the value toward 1.
Q11 · ARI-FRC-048 — Answer: E
Step 1 — convert both mixed numbers to improper fractions using a b/c = (ac + b)/c. 3 1/3 = (3 x 3 + 1)/3 = 10/3. 2 1/4 = (2 x 4 + 1)/4 = 9/4. Step 2 — cancel before multiplying: 10/3 x 9/4. The 3 divides into 9 (leaving 3), and the 2 in 10 cancels with the 4 (leaving 5 and 2). 10/3 x 9/4 = (5 x 3)/2 = 15/2. Step 3 — convert back: 15/2 = 7 1/2. Check in decimals: 3.333 x 2.25 = 7.5. Correct. Answer: 7 1/2.
Trap. Choice B, 6 1/12, is the classic error: multiplying the whole parts (3 x 2 = 6) and the fraction parts (1/3 x 1/4 = 1/12) separately and gluing them together. Mixed numbers are sums, not products, so that step drops the cross terms entirely. Choice D, 6 3/4, is 3 x 2 1/4 - the 1/3 was forgotten. Choice C, 6 1/4, keeps only the second fraction part. Choice A, 5 7/12, is the SUM 10/3 + 9/4 instead of the product.
Q12 · ARI-FRC-049 — Answer: 5/6
Step 1 — translate. 'What fraction of 3/4 is 5/8' means x x (3/4) = 5/8, where x is what we want. Step 2 — the quantity after 'of' is the one you divide BY: x = (5/8) / (3/4). Step 3 — flip the divisor and multiply: 5/8 x 4/3 = 20/24 = 5/6. Check: 5/6 of 3/4 = 5/6 x 3/4 = 15/24 = 5/8. Correct. Answer: 5/6.
Trap. Answering 5/8, by reading 'what fraction ... is 5/8' as 'what is 5/8'. The other error is dividing the wrong way round: (3/4)/(5/8) = 6/5, which is bigger than 1 and so cannot be 'a fraction OF 3/4' when 5/8 is smaller than 3/4. Whatever follows 'of' goes in the denominator.
Q13 · ARI-FRC-050 — Answer: A, B, C, E
Write x = n/d. Since 0 < n < d, we have 0 < x < 1. A - x^2 = x times x, and multiplying a positive number by a factor below 1 shrinks it, so x^2 < x. TRUE. B - d/n is the reciprocal of x. Since 0 < x < 1, its reciprocal exceeds 1. With d = 5, n = 2: 5/2 = 2.5 > 1. TRUE. C - cross-multiply (n+1)/(d+1) against n/d: d(n+1) vs n(d+1), i.e. dn + d vs dn + n. Cancelling dn leaves d vs n, and d > n, so (n+1)/(d+1) is larger. TRUE. D - for 0 < x < 1 the square root is LARGER, not smaller: sqrt(1/4) = 1/2 > 1/4. FALSE. E - x^3 = x^2 times x, again shrinking a positive number by a factor below 1, so x^3 < x^2. TRUE. Answer: A, B, C, E.
Trap. D is the planted reversal. Students correctly learn that squaring a proper fraction shrinks it, then assume taking a root must shrink it too - but a root and a square move in OPPOSITE directions, so sqrt(x) > x on (0, 1). C is the one people wrongly reject: adding 1 to both top and bottom pulls a proper fraction toward 1, which for a fraction below 1 means UP. Compare 2/3 < 3/4 < 4/5.
Q14 · ARI-FRC-051 — Answer: C
Work from the innermost bracket outward - never from the outside in. Step 1 — innermost: 1 + 1/2 = 3/2. Step 2 — the middle reciprocal: 1/(3/2) = 2/3. Step 3 — add the 1 outside it: 1 + 2/3 = 5/3. Step 4 — the outermost reciprocal: 1/(5/3) = 3/5. Check in decimals: 1 + 1/2 = 1.5; 1/1.5 = 0.6667; 1 + 0.6667 = 1.6667; 1/1.6667 = 0.6 = 3/5. Correct. Answer: 3/5.
Trap. Every distractor is a legitimate intermediate value, which is why this question punishes stopping early: B (3/2) is step 1, A (2/3) is step 2, D (5/3) is step 3. Choice E, 2/5, is the sign of a real error - substituting the 3/2 in place of the 1/(3/2), i.e. 1/(1 + 3/2) = 1/(5/2) = 2/5. That skips the middle reciprocal entirely.
Q15 · ARI-FRC-052 — Answer: A
Step 1 — factor the denominator of Column B as a difference of squares: 1 - x^2 = (1 - x)(1 + x). Step 2 — split Column B to expose Column A inside it: x^2/((1 - x)(1 + x)) = [x/(1 - x)] x [x/(1 + x)]. So Column B = Column A x [x/(1 + x)]. Step 3 — bound the extra factor. For 0 < x < 1 we have x < 1 + x, so 0 < x/(1 + x) < 1. Step 4 — Column A is positive (numerator x > 0, denominator 1 - x > 0), and multiplying a positive number by a factor strictly below 1 makes it smaller. So Column B < Column A. Check with x = 1/2: Column A = (1/2)/(1/2) = 1; Column B = (1/4)/(3/4) = 1/3. Check with x = 9/10: Column A = 9; Column B = 0.81/0.19 = 4.26. Column A wins both. Column A is greater.
Trap. Reading Column B as 'Column A squared' and, since squaring shrinks numbers below 1, guessing the answer flips somewhere - it does not. The real relationship is Column B = Column A x x/(1 + x), and that multiplier is below 1 for every allowed x, so Column A wins on the whole interval. Choosing D because one test value gave Column A = 1 (a value that could go either way against Column B) is the other slip; the algebra covers all x at once.
Q16 · ARI-FRC-053 — Answer: 1/7
Step 1 — name it and shift by the length of the repeating block. Let x = 0.142857142857... The block is 6 digits long, so multiply by 10^6: 1000000x = 142857.142857142857... Step 2 — subtract the original. The infinite tails are identical and cancel exactly: 1000000x - x = 142857 999999x = 142857 Step 3 — solve and reduce: x = 142857/999999. Note that 7 x 142857 = 999999, so the fraction is exactly 1/7. Check: 1/7 = 0.142857142857... Correct. Answer: 1/7. Shortcut worth memorising: a purely repeating block of k digits over k nines gives the fraction directly - 142857/999999 here, 25/99 for 0.252525..., 4/9 for 0.444...
Trap. Leaving the answer as 142857/999999, which is right but not in lowest terms and would be marked wrong. The other error is shifting by the wrong power: multiplying by 10 instead of 10^6 gives 10x = 1.42857142857..., and subtracting leaves 9x = 1.285714..., a decimal that still recurs, so nothing was gained. Shift by exactly the block length.
Q17 · ARI-FRC-054 — Answer: A
Set the common value to k, so p/q = r/s = k. Then p = kq and r = ks. A - (p + r)/(q + s) = (kq + ks)/(q + s) = k(q + s)/(q + s) = k. The (q + s) cancels for every choice of q and s, so this always equals p/q. TRUE. B - (p + q)/(r + s) = (kq + q)/(ks + s) = q(k + 1)/(s(k + 1)) = q/s. That is q/s, not k. With k = 7/3, q = 5, s = 2 (so p = 35/3, r = 14/3), it gives 5/2, not 7/3. FALSE. C - (p + s)/(q + r) pairs the terms across the two ratios. Same numbers: (35/3 + 2)/(5 + 14/3) = (41/3)/(29/3) = 41/29, not 7/3. FALSE. D - p^2/q^2 = k^2, which equals k only when k = 1. FALSE in general. E - (p + 1)/(q + 1) = (35/3 + 1)/(5 + 1) = (38/3)/6 = 19/9, not 7/3. Adding 1 to top and bottom drags the fraction toward 1. FALSE. Answer: A.
Trap. Choice E is the most tempting because A and E look like the same move. They are not: in A the numerator gains r and the denominator gains s, and r/s is ITSELF equal to k, so the added piece is already in the right proportion. In E the added piece is 1/1, a ratio of 1, which pulls the value toward 1. The rule to keep: you may add numerators to numerators and denominators to denominators only when the two ratios are equal.
Q18 · ARI-FRC-057 — Answer: 32
The insight: never solve for a and b separately. Combine the reciprocals over a common denominator and the two given facts line up. Step 1 — 1/a + 1/b = b/(ab) + a/(ab) = (a + b)/(ab). Step 2 — substitute the given product: (a + b)/192 = 1/6. Step 3 — cross-multiply: 6(a + b) = 192, so a + b = 32. Check by finding the actual pair: a = 8 and b = 24 give ab = 192 and 1/8 + 1/24 = 3/24 + 1/24 = 4/24 = 1/6. And 8 + 24 = 32. Correct. Answer: 32.
Trap. Setting up the quadratic and grinding for a and b individually - a longer road to the same place, and easy to abandon halfway. The arithmetic error to watch for is 1/a + 1/b = 1/(a + b), which would give a + b = 6; that identity is false, as 1/8 + 1/24 = 1/6 while 1/(8 + 24) = 1/32 shows. The correct identity is (a + b)/ab.
Q19 · ARI-FRC-058 — Answer: C
The expression is the square of a sum minus the sum of the squares. They are different numbers, so compute each piece separately. Step 1 — factor 0.1 out of the first bracket: 0.1 + 0.2 + ... + 0.9 = 0.1 x (1 + 2 + ... + 9). Step 2 — the integers 1 through 9 sum to 9 x 10/2 = 45, so the bracket is 0.1 x 45 = 4.5, and its square is 4.5^2 = 20.25. Step 3 — factor 0.01 out of the second bracket: 0.1^2 + 0.2^2 + ... + 0.9^2 = 0.01 x (1^2 + 2^2 + ... + 9^2). Step 4 — the squares 1 through 9 sum to n(n + 1)(2n + 1)/6 with n = 9: 9 x 10 x 19/6 = 285. So the second bracket is 0.01 x 285 = 2.85. Step 5 — subtract: 20.25 - 2.85 = 17.4. Check the place count: 45^2 = 2025 with two decimal places gives 20.25, and 285 with two decimal places gives 2.85; 20.25 - 2.85 = 17.4. Correct. Answer: 17.4.
Trap. Choice A, 0, is the belief that the square of a sum equals the sum of the squares, which would make the two brackets cancel; (a + b)^2 = a^2 + 2ab + b^2, and here the cross terms are worth 17.4. Choice D, 20.25, is the first bracket alone, subtracting nothing. Choice B, 1.65, forgets to square the first sum: 4.5 - 2.85. Choice E, 26.4, runs the series to 1.0 instead of stopping at 0.9, giving 5.5^2 - 3.85 = 30.25 - 3.85.
Q20 · ARI-FRC-062 — Answer: B
Step 1 — split each term with partial fractions. For every n, 1/(n(n+1)) = 1/n - 1/(n+1). Verify on the first term: 1/1 - 1/2 = 1/2, and 1/(1 x 2) = 1/2. Correct. Step 2 — write the whole sum out that way: (1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/9 - 1/10) Step 3 — every interior term appears once positive and once negative and cancels. Only the very first and the very last survive: Sum = 1/1 - 1/10 = 9/10. Check by brute force on a short version: 1/2 + 1/6 + 1/12 = 6/12 + 2/12 + 1/12 = 9/12 = 3/4, and the telescoped form gives 1 - 1/4 = 3/4. Correct. Answer: 9/10.
Trap. Choice C, 1, is telescoping correctly but forgetting to subtract the final -1/10. Choice D, 1/10, keeps only that discarded end term. Choice A, 1/2, is stopping after the first term. Choice E, 10/9, is the answer inverted. The lesson: after telescoping, always name which two ends survive - here the leading 1/1 and the trailing -1/10.
Q21 · ARI-FRC-064 — Answer: B
Step 1 — use the standard recurring-decimal method. Let x = 0.999... The repeating block is one digit long, so multiply by 10: 10x = 9.999... Step 2 — subtract. The infinite tails are identical and cancel exactly: 10x - x = 9.999... - 0.999... = 9 9x = 9 x = 1. Step 3 — a second confirmation: 1/3 = 0.333... exactly. Multiply both sides by 3: 1 = 0.999... Step 4 — a third: if x were less than 1, there would have to be a positive gap 1 - x. But 1 - 0.999... is smaller than 0.1, smaller than 0.01, smaller than every positive number, so the gap is 0. Answer: x = 1.
Trap. Choice A is picked by almost everyone, on the feeling that 0.999... is 'creeping up on 1 without ever arriving'. There is no arriving to do - 0.999... is not a process, it is a single number, and that number is 1. Choices D and E fail for the same reason: x = 1 is an integer, so it is rational and is trivially a ratio of integers.
Q22 · ARI-FRC-065 — Answer: 1/9
Step 1 — chain the ratios. The interior letters cancel in a telescoping product: (a/b) x (b/c) x (c/d) x (d/e) = a/e. Step 2 — substitute: a/e = (3/4) x (2/5) x (5/6) x (4/9). Step 3 — cancel before multiplying. The 5 in 2/5 cancels the 5 in 5/6; the 4 in 3/4 cancels the 4 in 4/9: (3/4) x (2/5) x (5/6) x (4/9) = (3 x 2)/(6 x 9) = 6/54 = 1/9. Check with actual numbers. Build forward from a = 3: b = 4, then c = 10 (since 4/c = 2/5), then d = 12 (since 10/d = 5/6), then e = 27 (since 12/e = 4/9). All four given ratios hold, and a/e = 3/27 = 1/9. Correct. Answer: 1/9.
Trap. Multiplying only the first and last ratios, (3/4) x (4/9) = 1/3, on the assumption that the middle links 'cancel themselves'. They do not - each middle ratio contributes a real factor, and here 2/5 x 5/6 = 1/3 is exactly what turns 1/3 into 1/9. The other error is adding the four fractions instead of multiplying them.
Q23 · ARI-FRC-066 — Answer: B, C, E
The rule: reduce to lowest terms FIRST, then factor the denominator. It terminates if and only if the reduced denominator's prime factorisation contains nothing but 2s and 5s. A - 3/14 is already reduced; 14 = 2 x 7. The 7 forces recurrence. FALSE. (3/14 = 0.2142857142857...) B - 7/8 is reduced; 8 = 2^3, only 2s. TRUE. (0.875) C - 18/48 looks bad because 48 = 2^4 x 3, but it is not reduced: divide top and bottom by 6 to get 3/8, and 8 = 2^3. TRUE. (0.375) D - 11/12 is reduced; 12 = 2^2 x 3. The 3 forces recurrence. FALSE. (0.91666...) E - 21/35 looks bad because 35 = 5 x 7, but divide top and bottom by 7 to get 3/5. TRUE. (0.6) Answer: B, C, E.
Trap. Judging the denominator as printed. C and E both carry a forbidden prime (3 in 48, 7 in 35) that vanishes on reducing, so a student who factors before cancelling rejects both and answers B alone. The mirror error would be accepting a fraction whose reduced denominator still hides a 3 or 7 - which is why A and D must be checked in reduced form too. Reduce first, factor second.
Q24 · ARI-FRC-068 — Answer: D
Step 1 — turn each time into a rate. Pipe A fills 1/6 of the tank per hour; pipe B fills 1/4 per hour. Step 2 — rates add, so the combined rate is 1/6 + 1/4. LCD is 12: 2/12 + 3/12 = 5/12 of the tank per hour. Step 3 — multiply rate by time, writing 1.5 as 3/2: (5/12) x (3/2) = 15/24 = 5/8. Check: in 1.5 hours pipe A contributes 1.5/6 = 1/4 and pipe B contributes 1.5/4 = 3/8. Then 1/4 + 3/8 = 2/8 + 3/8 = 5/8. Correct. Answer: 5/8.
Trap. Choice A, 1/5, is adding the rates across: 1/6 + 1/4 = 2/10. Choice B, 3/10, carries that same wrong rate through the 1.5 hours. Choice C, 5/12, is the correct combined rate but reported as the answer - it is the fraction filled in ONE hour, not in 1.5. Choice E, 5/6, rounds 1.5 hours up to 2. Times never add in work problems; rates do.
Q25 · ARI-FRC-070 — Answer: A, B, E
For a purely repeating decimal, put the repeating block over as many 9s as the block has digits, then reduce. A - one repeating digit: 1/9. Directly: x = 0.111..., 10x = 1.111..., 9x = 1, x = 1/9. TRUE. B - two repeating digits: 12/99. Divide top and bottom by 3: 12/99 = 4/33. TRUE. C - two repeating digits: 36/99 = 4/11 = 0.363636... But 9/25 = 0.36 exactly, a TERMINATING decimal that stops after two places. 4/11 is not 9/25. FALSE. D - here the 1 does NOT repeat; only the 6 does, so the nines rule does not apply as written. 16/99 = 0.161616..., which repeats '16'. The true value is 0.1666... = 1/6. FALSE. E - two repeating digits: 45/99. Divide top and bottom by 9: 45/99 = 5/11. TRUE. Answer: A, B, E.
Trap. C exploits the gap between 0.36 and 0.363636... - the second is bigger, and no terminating fraction can equal a genuinely recurring one. D exploits the mixed decimal 0.1666..., where one digit sits OUTSIDE the repeating block; sweeping both digits into a two-digit block gives 0.161616..., a different number. Before applying the nines rule, check that the repetition starts immediately after the decimal point.
Q26 · ARI-FRC-069 — Answer: 11/20
Multiplying nine fractions directly is hopeless by hand. Factor each one instead. Step 1 — difference of squares on every factor: 1 - 1/n^2 = (n^2 - 1)/n^2 = ((n - 1)(n + 1))/(n x n) = ((n-1)/n) x ((n+1)/n). Step 2 — split the whole product into two chains: Product = [(1/2)(2/3)(3/4)...(9/10)] x [(3/2)(4/3)(5/4)...(11/10)]. Step 3 — each chain telescopes. In the first, every numerator cancels the previous denominator, leaving 1/10. In the second, the same happens, leaving 11/2. Step 4 — multiply the survivors: (1/10) x (11/2) = 11/20. Check the pattern on a short case: for n = 2 and 3 only, (3/4)(8/9) = 24/36 = 2/3, and the formula (N+1)/(2N) with N = 3 gives 4/6 = 2/3. Correct. With N = 10 the formula gives 11/20. Answer: 11/20.
Trap. Grinding out (3/4)(8/9)(15/16)... term by term. The numerators reach 3 x 8 x 15 x 24 x ... and the arithmetic collapses long before the ninth factor. The specific slip once the split is found is losing track of the surviving ends: the first chain leaves the FIRST numerator (1) over the LAST denominator (10), while the second leaves the LAST numerator (11) over the FIRST denominator (2). Getting either end wrong turns 11/20 into 1/20 or 11/2.
Q27 · ARI-FRC-072 — Answer: C
Step 1 — name the two-digit block. As a number, the block 'ab' equals 10a + b; call it N, so 0 < N < 100. Step 2 — shift by the block length. Since the block is 2 digits, multiply x by 100: 100x = N.ababab... Step 3 — the tail after the decimal point in 100x is identical to x itself, so 100x = N + x. Step 4 — solve: 99x = N, hence x = N/99 = (10a + b)/99. That is exactly Column B, for every allowed pair of digits. The two quantities are equal. Check with a = 1, b = 2: x = 0.121212... and (10 + 2)/99 = 12/99 = 4/33 = 0.121212... Check with a = 9, b = 0: x = 0.909090... and 90/99 = 10/11 = 0.909090... Both agree.
Trap. Choosing D on the grounds that a and b are unknown. They never need to be known: the shift-and-subtract works symbolically, and the digits cancel out of the comparison entirely. The other route to D is testing a single case such as a = b = 3 (x = 0.3333... = 33/99 = 1/3), getting equality, and then doubting whether it survives other digits - it does, because Step 4 used nothing about a and b beyond their being the repeating block.
Q28 · ARI-FRC-077 — Answer: 1/20
Computing 19 steps one at a time is not the intended route - find the pattern, then prove it. Step 1 — generate the first few terms. a(1) = 1 a(2) = 1/(1 + 1) = 1/2 a(3) = (1/2)/(1 + 1/2) = (1/2)/(3/2) = (1/2) x (2/3) = 1/3 a(4) = (1/3)/(1 + 1/3) = (1/3)/(4/3) = (1/3) x (3/4) = 1/4 Step 2 — conjecture a(n) = 1/n. Step 3 — prove it carries forward. If a(n-1) = 1/(n-1), then a(n) = [1/(n-1)] / (1 + 1/(n-1)) = [1/(n-1)] / [n/(n-1)] = [1/(n-1)] x [(n-1)/n] = 1/n. So the pattern holds at every step, and a(1) = 1/1 starts it off. Step 4 — a(20) = 1/20. A slicker route: take reciprocals. If b(n) = 1/a(n), the rule becomes b(n) = (1 + a(n-1))/a(n-1) = 1/a(n-1) + 1 = b(n-1) + 1. So the reciprocals just count up: 1, 2, 3, ... and b(20) = 20, giving a(20) = 1/20. Answer: 1/20.
Trap. Grinding term by term and slipping on the complex fraction at some point - each step needs (1/k)/((k+1)/k), and dividing instead of flipping turns 1/(k+1) into 1/(k(k+1)), after which every later term is wrong. The reciprocal substitution removes the fractions entirely and shows why the answer is simply 1/n: adding 1 to a number's reciprocal is what the rule does, once you look at it upside down.
Q29 · ARI-FRC-078 — Answer: A, C, D
Strip any factors of 2 and 5 from the denominator - they only shift the decimal point and never join the repeating block. Whatever is left determines the period. A - 1/7 = 0.142857 142857 ... The block 142857 has 6 digits. TRUE. B - 1/11 = 0.09 09 09 ... The repeating block is 09, only 2 digits long. FALSE. C - 1/13 = 0.076923 076923 ... The block 076923 has 6 digits. TRUE. D - 1/14: since 14 = 2 x 7, write 1/14 = (1/2) x (1/7) = 0.0714285 714285 ... The leading 0 is the shift caused by the factor 2; the repeating part is 714285, the same six digits as 1/7 rotated. The period is inherited from the 7. TRUE. E - 1/9 = 0.111... The block is 1, a single digit. FALSE. Answer: A, C, D.
Trap. Rejecting D. The 2 in 14 = 2 x 7 makes the denominator look unlike 7, but a factor of 2 or 5 can only delay the start of the repetition, never change its length - the period of 1/14 is the period of 1/7. The other miss is assuming any denominator that is not 2 or 5 gives a long block: 1/11 has period 2 and 1/9 has period 1, so the block length has to be checked, not guessed from the size of the denominator.
Q30 · ARI-FRC-080 — Answer: A
The two columns are close, so an estimate will not settle it - the sum must be computed exactly. Step 1 — factor and split each term. n^2 - 1 = (n - 1)(n + 1), and 1/((n-1)(n+1)) = (1/2)[1/(n-1) - 1/(n+1)]. The 1/2 appears because the two parts are 2 apart, not 1 apart. Verify at n = 2: (1/2)[1/1 - 1/3] = (1/2)(2/3) = 1/3, and 1/(2^2 - 1) = 1/3. Correct. Step 2 — write the sum out. Pulling the 1/2 in front: Sum = (1/2)[(1/1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + ... + (1/9 - 1/11)] Step 3 — because each negative term is two places ahead, cancellation leaves TWO terms at each end, not one. The positives 1/1 and 1/2 are never cancelled, and the negatives -1/10 and -1/11 are never cancelled. Sum = (1/2)[1 + 1/2 - 1/10 - 1/11] Step 4 — evaluate the bracket over 110: 110/110 + 55/110 - 11/110 - 10/110 = 144/110. Sum = (1/2)(144/110) = 72/110 = 36/55. Step 5 — compare 36/55 with 13/20 by cross-multiplication: 36 x 20 = 720 versus 13 x 55 = 715. Since 720 > 715, Column A is greater. Column A = 36/55 = 0.6545..., Column B = 0.65. Column A is greater.
Trap. Two traps stack. First, forgetting the factor of 1/2 in the partial fraction: 1/((n-1)(n+1)) is NOT 1/(n-1) - 1/(n+1), which would double the answer to 72/55. Second, assuming only one term survives at each end, as in the familiar 1/(n(n+1)) telescope - here the gap of 2 means two terms survive at each end, and dropping the 1/2 and the -1/11 gives (1/2)(1 - 1/10) = 9/20 = 0.45, which would point the wrong way. The margin is only 720 versus 715, so estimating to two decimals and calling it equal (choice C) is exactly the wrong move.