AFOQT math: Arithmetic Reasoning and Math Knowledge

Twice a number, decreased by 7, equals 3 more than the number. What is the number?

1. 10
   Translate as 2n − 7 = n + 3; subtracting n and adding 7 gives n = 10 (check: 20 − 7 = 13 = 10 + 3).
2. −4
   −4 moves the 7 across without changing its sign (n = 3 − 7); adding 7 to both sides gives n = 10.
3. 5
   5 drops the n on the right side and solves 2n − 7 = 3.
4. 4/3
   4/3 writes "twice a number, decreased by 7" as 7 − 2n; 7 − 2n = n + 3 gives n = 4/3.
5. −7
   −7 reads "3 more than the number" as 3n: 2n − 7 = 3n gives n = −7.

ID 1821338 · Report a problem

A unit of 198 people has officers and enlisted members in a ratio of 2 to 9. How many officers are in the unit?

1. 162
   162 is 9 × 18, the number of enlisted members, not the number of officers.
2. 18
   18 is one part of the ratio (198 ÷ 11); the officers make up 2 parts, so they number 36.
3. 44
   44 is 2 × (198 ÷ 9): it uses 9 parts for the whole unit, but 2 + 9 = 11 parts make up the 198 people.
4. 36
   The ratio 2 to 9 has 11 parts, so one part is 198 ÷ 11 = 18 people and the officers are 2 × 18 = 36.
5. 22
   22 divides 198 by 9, as if the enlisted share were the whole unit, and stops there.

ID 1821351 · Report a problem

Adult tickets cost $12 and child tickets cost $7. A group of 9 people paid $88 in all. How many children were in the group?

1. 2
   2 divides the extra $25 above an all-child price ($88 − $63) by $12; that extra must be divided by the $5 price difference.
2. 3
   3 rounds $20 ÷ $7: it divides the $20 savings by the child price instead of by the $5 difference between the prices.
3. 4
   Nine adults would cost $108; each child saves $5, and the group saved $20, so there were 20 ÷ 5 = 4 children (5 × $12 + 4 × $7 = $88).
4. 7
   7 is how many $12 tickets fit into $88 (88 ÷ 12 ≈ 7.3); that ignores that the group has 9 people of two kinds.
5. 5
   5 is the number of adults in the group; the question asks for children.

ID 1821383 · Report a problem

Crates are shared among three squadrons in the ratio 2:3:4. The largest share is 72 crates more than the smallest share. How many crates are there in all?

1. 288
   288 is 72 × 4, the 72-crate gap multiplied by the largest ratio term.
2. 324
   The largest and smallest shares differ by 4 − 2 = 2 parts, so one part is 72 ÷ 2 = 36 crates and all 9 parts are 324.
3. 216
   216 multiplies 72 by the 3 squadrons; the 72 crates are a difference of 2 parts, not one squadron's share.
4. 162
   162 treats the 72-crate gap as 4 parts (one part = 18); the gap between 4 parts and 2 parts is only 2 parts.
5. 144
   144 is the largest share alone (4 parts × 36), not the total of all three shares.

ID 1821392 · Report a problem

A supply officer is cataloging equipment costs. A radio (R) costs $50 more than twice the cost of a headset (H). A battery (B) costs one-third as much as a headset. Which equation gives the cost of a radio in terms of the cost of a battery?

1. R = 5B + 50
   R = 5B + 50 adds the factors 2 and 3 instead of multiplying them.
2. R = 6B + 50
   From B = H/3, a headset costs H = 3B; substituting into R = 2H + 50 gives R = 2(3B) + 50, so R = 6B + 50.
3. R = 6B + 150
   R = 6B + 150 multiplies the $50 by 3 as well; only H is replaced by 3B.
4. R = (2/3)B + 50
   R = (2/3)B + 50 reads the battery relation backward, as H = B/3.
5. R = 2B + 50
   R = 2B + 50 puts B in place of H without converting; a headset costs three batteries.

ID 1858829 · Report a problem

Two aircraft leave the same airfield at noon and fly in opposite directions, one at 45 knots and the other at 75 knots. At what time are they 360 nautical miles apart?

1. 8:00 p.m.
   8:00 p.m. uses only the slower aircraft (360 ÷ 45 = 8 hours).
2. 12:00 midnight
   12:00 midnight subtracts the speeds (360 ÷ 30 = 12 hours), which fits two aircraft flying the same way, not opposite ways.
3. 6:00 p.m.
   6:00 p.m. uses the average speed of 60 knots for a single aircraft; the distances of the two aircraft add, so the speeds add.
4. 3:00 p.m.
   Flying in opposite directions, the gap grows at 45 + 75 = 120 knots, so 360 ÷ 120 = 3 hours after noon, 3:00 p.m.
5. 4:48 p.m.
   4:48 p.m. uses only the faster aircraft (360 ÷ 75 = 4.8 hours) and ignores the slower one moving away too.

ID 1941509 · Report a problem

A club of 32 members collects $15 in dues from each member. It pays $180 to rent a hall and $6.50 per member for food. How much money is left?

1. $272.00
   $272.00 subtracts the food but forgets the $180 hall rental.
2. $300.00
   $300.00 subtracts the hall rental but forgets the $208 for food.
3. $98.50
   $98.50 pays for food for 31 members instead of 32 (an off-by-one slip of $6.50).
4. $293.50
   $293.50 subtracts $6.50 for food once instead of $6.50 for each of the 32 members.
5. $92.00
   Dues bring in 32 × $15 = $480; the hall costs $180 and food 32 × $6.50 = $208, so $480 − $180 − $208 = $92.00 is left.

ID 1941520 · Report a problem

A 10 by 8 by 6 block has a 2 by 2 square hole cut straight through its 10-unit length. What volume remains?

1. 440
   The block is 10 × 8 × 6 = 480, and the hole is a 2 × 2 × 10 prism of 40, so 480 − 40 = 440 remains.
2. 480
   480 is the volume of the solid block; the hole has not been taken out.
3. 448
   448 subtracts a hole 8 units long (2 × 2 × 8 = 32); the hole runs through the 10-unit length.
4. 456
   456 subtracts a hole 6 units long (2 × 2 × 6 = 24) instead of 10 units long.
5. 470
   470 subtracts only 10, the hole's length, instead of its volume 2 × 2 × 10 = 40.

ID 2009966 · Report a problem

If 2x/5 − 14 = 6, then x =

1. 20
   20 is the value of 2x/5 after adding 14; the last step, solving for x, is missing.
2. 22
   22 multiplies 2x/5 by 5 but not the 14: 2x − 14 = 30 gives x = 22. Every term has to be multiplied.
3. −20
   −20 subtracts 14 from 6 instead of adding it (2x/5 = −8).
4. 8
   8 multiplies 20 by 2/5 instead of by its reciprocal 5/2.
5. 50
   Add 14 to get 2x/5 = 20, then multiply by 5/2: x = 50 (check: 2 × 50 / 5 − 14 = 6).

ID 1821294 · Report a problem

If x + y = 50 and 0.6x + 0.2y = 22, then x =

1. 12
   12 is the value of 0.4x; dividing by 0.4 is the missing last step.
2. 15
   15 writes 0.6x + 0.2(50 − x) as 0.6x + 0.2x + 10, adding the coefficients, so 0.8x = 12.
3. 30
   Substitute y = 50 − x: 0.6x + 10 − 0.2x = 22, so 0.4x = 12 and x = 30 (then y = 20, and 18 + 4 = 22).
4. 20
   20 is the value of y, not x.
5. 25
   25 just splits 50 in half without using the second equation.

ID 1821298 · Report a problem

What is the volume of a rectangular box 4 by 5 by 6?

1. 148
   148 is the surface area, 2 × (20 + 24 + 30), not the volume.
2. 74
   74 adds the three face areas once (20 + 24 + 30), half the surface area; it is not a volume.
3. 30
   30 is the area of one face, 5 × 6; the third dimension is missing.
4. 15
   15 adds the three dimensions (4 + 5 + 6) instead of multiplying them.
5. 120
   Volume = length × width × height = 4 × 5 × 6 = 120.

ID 1821346 · Report a problem

If 3u + v = 7 and u − v = 1, then

1. u = 1, v = 4
   u = 1, v = 4 satisfies 3u + v = 7 but gives u − v = −3, not 1.
2. u = 3, v = 2
   u = 3, v = 2 satisfies u − v = 1 but gives 3u + v = 11.
3. u = 4, v = 3
   u = 4, v = 3 satisfies only u − v = 1; 3u + v would be 15.
4. u = 2, v = 1
   Adding the equations removes v: 4u = 8, so u = 2, v = 1; the pair satisfies both (6 + 1 = 7, 2 − 1 = 1).
5. u = 0, v = 7
   u = 0, v = 7 satisfies only the first equation; u − v would be −7.

ID 1821413 · Report a problem

For x, y > 0, √(72x⁵y⁸) equals

1. 6x²y⁴√(2x)
   72 = 36 × 2, x⁵ = x⁴ · x and y⁸ = (y⁴)²; the square roots 6, x², y⁴ come out and 2x stays inside: 6x²y⁴√(2x).
2. 6x²y²√(2x)
   6x²y²√(2x) takes the fourth root of y⁸ instead of the square root; √(y⁸) = y⁴.
3. 6x⁴y⁴√(2x)
   6x⁴y⁴√(2x) brings x⁴ out without halving its exponent; √(x⁴) = x².
4. 6x²y⁴√2
   6x²y⁴√2 loses the leftover x: x⁵ = x⁴ · x, and that single x stays under the root.
5. 36x²y⁴√(2x)
   36x²y⁴√(2x) brings 36 out without taking its square root; √36 = 6.

ID 1858819 · Report a problem

A tool listed at $250 is sold for $195. What percent discount was given?

1. 22%
   The discount is $250 − $195 = $55, and 55 ÷ 250 = 0.22, a 22% discount off the list price.
2. 28%
   28% divides the $55 discount by the $195 sale price; a discount is a share of the list price.
3. 2.2%
   2.2% misplaces the decimal point when turning 0.22 into a percent.
4. 78%
   78% is 195 ÷ 250, the share of the list price that was paid, not the discount.
5. 55%
   55% copies the $55 dollar discount as if it were a percent.

ID 1940693 · Report a problem

0.375 written as a fraction in lowest terms is

1. 37/100
   37/100 = 0.37 drops the last digit, 5 thousandths.
2. 5/8
   5/8 = 0.625 is the complement, 1 − 0.375, not 0.375 itself.
3. 3/8
   0.375 = 375/1000, and dividing both by 125 gives 3/8 (3 ÷ 8 = 0.375).
4. 15/4
   15/4 = 3.75 puts 375 over 100; three decimal places mean a denominator of 1,000.
5. 3/80
   3/80 = 0.0375, one place value too small.

ID 1940759 · Report a problem

A cube has a surface area of 54. What is its volume?

1. 54
   54 is the surface area; volume and surface area of a cube are different quantities.
2. 27
   Each of the 6 faces has area 54 ÷ 6 = 9, so an edge is 3 and the volume is 3³ = 27.
3. 9
   9 is the area of one face; the edge is √9 = 3 and the volume is 3³.
4. 81
   81 squares the face area 9 instead of cubing the edge 3.
5. 729
   729 cubes the face area 9 instead of the edge length 3.

ID 1941448 · Report a problem