AR · Lesson test

Fractions and Mixed Numbers Lesson Test

Ready to find out whether your fraction skills hold up? This lesson test gives you 15 original practice questions in the style of the ASVAB Arithmetic Reasoning subtest, all built around the Fractions and Mixed Numbers lesson. Expect items on simplifying, comparing, converting mixed numbers, the four operations and fraction-based word problems.

15-question lesson test
35 more practice questions
Every answer explained

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Question 1 of 15 · easy

A repair crew has finished 24 of its 40 assigned work orders. What fraction of the work orders is finished, in simplest form?

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Answer: A

Correct answer: 3/5. The finished part over the whole is $\frac{24}{40}$ . The greatest common factor is 8, so

\frac{24 \div 8}{40 \div 8} = \frac{3}{5}

A (correct): 24 out of 40 is 24/40, and dividing top and bottom by 8 gives 3/5.
B: This turns the fraction upside down, putting the whole over the part.
C: This is the fraction not yet finished, 16 out of 40.
D: This compares finished orders to unfinished ones (24 to 16), not finished to the whole.

Question 2 of 15 · easy

Which statement about $\frac{5}{8}$ and $\frac{7}{11}$ is true?

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Answer: A

Correct answer: 7/11 is greater. Cross-multiply:

5 \times 11 = 55 and 7 \times 8 = 56

The larger product, 56, belongs with 7/11, so $\frac{7}{11}$ is greater.

A (correct): Cross-multiplying gives 7 x 8 = 56, which beats 5 x 11 = 55, so 7/11 is greater.
B: Cross-multiplying gives 5 x 11 = 55 and 7 x 8 = 56, so 5/8 is the smaller fraction.
C: The cross products 55 and 56 differ, so the fractions do not have the same value.
D: Two fractions with known numerators and denominators can always be compared, for example by cross-multiplying.

Question 3 of 15 · easy

Which improper fraction is equal to $3 \frac{2}{7}$ ?

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Answer: D

Correct answer: 23/7. Multiply the whole number by the denominator and add the numerator:

3 \times 7 + 2 = 23

Keep the denominator:

3 \frac{2}{7} = \frac{23}{7}

A: This adds the whole number to the numerator without first multiplying by the denominator.
B: This multiplies the whole number by the numerator (3 x 2) and then adds the denominator.
C: This multiplies the whole number by the fraction instead of adding them.
D (correct): Multiply the whole number by the denominator and add the numerator: 3 x 7 + 2 = 23, over 7.

Question 4 of 15 · easy

A storeroom holds 64 batteries. If $\frac{3}{8}$ of them are issued to a patrol, how many batteries are issued?

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Answer: A

Correct answer: 24 batteries. \"Of\" means multiply. Divide by the denominator, then multiply by the numerator:

64 \div 8 = 8 and 8 \times 3 = 24

batteries.

A (correct): 3/8 of 64 means 64 / 8 = 8, then 8 x 3 = 24.
B: This is the 5/8 that stay in the storeroom, not the 3/8 that are issued.
C: This multiplies by 3 but never divides by 8, giving more batteries than the storeroom holds.
D: This divides by the denominator but stops before multiplying by the numerator.

Question 5 of 15 · easy

What is $\frac{1}{4} + \frac{2}{3}$ ?

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Answer: A

Correct answer: 11/12. Use the common denominator 12:

\frac{1}{4} = \frac{3}{12} and \frac{2}{3} = \frac{8}{12} then \frac{3}{12} + \frac{8}{12} = \frac{11}{12}

A (correct): Renaming as twelfths gives 3/12 + 8/12 = 11/12.
B: This subtracts 1/4 from 2/3 instead of adding.
C: This adds the numerators and the denominators, but fourths and thirds are different-sized parts.
D: This multiplies the fractions instead of adding them.

Question 6 of 15 · medium

What is $6 \frac{1}{3} - 2 \frac{3}{4}$ ?

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Answer: C

Correct answer: 3 7/12. Rename with twelfths: $6 \frac{4}{12} - 2 \frac{9}{12}$ . Since 4/12 is smaller than 9/12, borrow 1:

5 \frac{16}{12} - 2 \frac{9}{12} = 3 \frac{7}{12}

A: This adds 12/12 to the fraction when borrowing but forgets to lower the whole number from 6 to 5.
B: This borrows from the whole number but then subtracts the fractions in the wrong order.
C (correct): Rewriting 6 4/12 as 5 16/12 and subtracting 2 9/12 gives 3 7/12.
D: This subtracts the smaller fraction from the larger one (9/12 - 4/12) instead of borrowing.

Question 7 of 15 · medium

A recipe for one batch of trail mix calls for $2 \frac{1}{2}$ cups of oats. How many cups of oats are needed for $1 \frac{1}{3}$ batches?

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Answer: D

Correct answer: 3 1/3 cups. Change both mixed numbers to improper fractions and multiply:

\frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3 \frac{1}{3} cups

A: This adds the two mixed numbers instead of multiplying them.
B: This divides 2 1/2 by 1 1/3 instead of multiplying.
C: This multiplies the whole parts and the fraction parts separately, which skips the cross terms.
D (correct): Converting first, 5/2 x 4/3 = 20/6 = 10/3, which is 3 1/3 cups.

Question 8 of 15 · medium

A 6-foot metal rod is cut into pieces that are each $\frac{3}{8}$ foot long. How many pieces are made?

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Answer: D

Correct answer: 16 pieces. \"How many pieces fit\" means divide. Multiply by the reciprocal of the divisor:

6 \div \frac{3}{8} = 6 \times \frac{8}{3} = 16

pieces.

A: This multiplies 6 by 3/8; cutting into pieces asks how many fit, which is division.
B: This multiplies 6 by 8, which counts eighths of a foot instead of 3/8-foot pieces.
C: This divides 3/8 by 6, reversing the order of the division.
D (correct): 6 / (3/8) = 6 x 8/3 = 48/3 = 16 pieces.

Question 9 of 15 · medium

A shelf board is $3 \frac{3}{8}$ feet long. How long is it in inches?

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Answer: B

Correct answer: 40 1/2 inches. Change to an improper fraction and multiply by 12 inches per foot:

\frac{27}{8} \times 12 = \frac{324}{8} = 40 \frac{1}{2} inches

A: This converts the 3 feet to inches but leaves the 3/8 as if it were already in inches.
B (correct): Since 3 3/8 = 27/8, multiplying by 12 gives 324/8 = 40 1/2 inches.
C: This multiplies the numerator 27 by 12 but forgets to divide by the denominator 8.
D: This reads 3/8 foot as 3 inches; 3/8 of 12 inches is 4 1/2 inches.

Question 10 of 15 · medium

A tool kit holds 54 wrenches and screwdrivers. The ratio of wrenches to screwdrivers is 4:5. How many wrenches are in the kit?

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Answer: C

Correct answer: 24 wrenches. The ratio 4:5 has 9 parts in all, so wrenches are $\frac{4}{9}$ of the kit:

54 \div 9 = 6 and 6 \times 4 = 24

wrenches.

A: This is the number of screwdrivers, which make up 5/9 of the kit.
B: This is the size of one ratio part; it still has to be multiplied by 4.
C (correct): The ratio has 4 + 5 = 9 parts, so wrenches are 4/9 of 54, which is 24.
D: This takes 4/5 of 54, using the second ratio number as the total number of parts.

Question 11 of 15 · medium

A section has a $1,200 budget. It spends $\frac{1}{4}$ of the budget on fuel and $\frac{2}{5}$ of the budget on spare parts. How many dollars of the budget are left?

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Answer: B

Correct answer: $420. Spent:

\frac{1}{4} + \frac{2}{5} = \frac{5}{20} + \frac{8}{20} = \frac{13}{20}

Left: $\frac{7}{20}$ of the budget, and

1, 200 \div 20 \times 7 = 420 dollars

A: This uses the difference 2/5 - 1/4 = 3/20 instead of the part left after both expenses.
B (correct): The spending is 1/4 + 2/5 = 13/20, leaving 7/20, and 7/20 of 1,200 is 420.
C: This is the amount spent, 13/20 of the budget, not the amount left.
D: This adds the fractions straight across to get 3/9 and finds the remaining 2/3 of the budget.

Question 12 of 15 · hard

A 60-gallon water tank is $\frac{2}{3}$ full. A crew then uses $\frac{3}{4}$ of the water that is in the tank. How many gallons are left in the tank?

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Answer: D

Correct answer: 10 gallons. Water in the tank:

\frac{2}{3} \times 60 = 40 gallons

Used:

\frac{3}{4} \times 40 = 30 gallons

Left:

40 - 30 = 10 gallons

which is also $\frac{1}{4}$ of 40.

A: This takes 1/4 of the full 60 gallons, ignoring that the tank started only 2/3 full.
B: This is the empty space in the tank before the crew used any water.
C: This is the amount of water the crew used, not the amount left.
D (correct): The tank holds 2/3 x 60 = 40 gallons, the crew uses 3/4 x 40 = 30, and 40 - 30 = 10 gallons remain.

Question 13 of 15 · hard

A hiker covers $2 \frac{1}{4}$ miles in $\frac{3}{4}$ hour. At that same steady pace, how long will it take her to hike $7 \frac{1}{2}$ miles?

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Answer: C

Correct answer: 2 1/2 hours. Speed:

\frac{9}{4} \div \frac{3}{4} = \frac{9}{4} \times \frac{4}{3} = 3 miles

per hour. Time:

7 \frac{1}{2} \div 3 = \frac{15}{2} \times \frac{1}{3} = \frac{5}{2} = 2 \frac{1}{2} hours

A: This multiplies the new distance by 3/4 hour, which does not give a time.
B: This multiplies the distance by the speed instead of dividing by it.
C (correct): Her speed is 9/4 / 3/4 = 3 mph, and 7 1/2 / 3 = 2 1/2 hours.
D: This treats 2 1/4 miles as the distance per hour, ignoring that it took only 3/4 hour.

Question 14 of 15 · hard

A cable $15 \frac{3}{4}$ feet long is cut into equal pieces, each $2 \frac{1}{4}$ feet long. How many pieces are made?

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Answer: B

Correct answer: 7 pieces. Convert to improper fractions:

15 \frac{3}{4} = \frac{63}{4} and 2 \frac{1}{4} = \frac{9}{4}

Divide by multiplying by the reciprocal:

\frac{63}{4} \times \frac{4}{9} = 7

pieces.

A: This divides the piece length by the cable length, reversing the division.
B (correct): Converting first, 63/4 / 9/4 = 63/4 x 4/9 = 7 pieces.
C: This multiplies the two lengths instead of dividing to see how many pieces fit.
D: This subtracts one piece length from the cable, which gives a length in feet, not a count.

Question 15 of 15 · hard

Which of these fractions has the greatest value?

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Answer: C

Correct answer: 4/5. Compare 4/5 with each of the others by cross-multiplying. Against 7/9:

4 \times 9 = 36 > 35 = 7 \times 5

Against 11/14:

56 > 55

Against 15/19:

4 \times 19 = 76 > 75 = 15 \times 5

So $\frac{4}{5}$ is greatest.

A: Against 4/5, the cross products are 75 and 76, so 15/19 is smaller.
B: Against 4/5, the cross products are 35 and 36, so 7/9 is smaller.
C (correct): Cross-multiplying against each other fraction, 4/5 always gives the larger product, for example 76 to 75 against 15/19.
D: Against 4/5, the cross products are 55 and 56, so 11/14 is smaller.

More practice

35 extra questions

Practice 1 of 35 · medium

Which list shows 2/3, 5/8 and 7/12 in order from least to greatest?

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Answer: C

Correct answer: 7/12, 5/8, 2/3. Rename all three over the common denominator 24.

\frac{2}{3} = \frac{16}{24}, \frac{5}{8} = \frac{15}{24}, \frac{7}{12} = \frac{14}{24}

Ordering the numerators 14, 15, 16 gives 7/12, 5/8, 2/3.

A: This is the correct order reversed, greatest to least, which happens when the larger denominator is assumed to mean the larger fraction.
B: This puts 5/8 below 7/12, but 5/8 is 15/24 while 7/12 is only 14/24.
C (correct): In twenty-fourths the fractions are 16/24, 15/24 and 14/24, so 7/12 is smallest and 2/3 is largest.
D: This places 2/3 below 5/8, but 2/3 is 16/24 and 5/8 is 15/24.

Practice 2 of 35 · medium

A plumber has a pipe that is $6 \frac{1}{8}$ feet long and cuts off a piece $2 \frac{5}{6}$ feet long. How long is the pipe that is left?

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Answer: D

Correct answer: 3 7/24 feet. Rename over 24: $6 \frac{3}{24} - 2 \frac{20}{24}$ . Because 3/24 is smaller than 20/24, borrow 1 whole to get $5 \frac{27}{24}$ .

Then

5 \frac{27}{24} - 2 \frac{20}{24} = 3 \frac{7}{24} feet

A: This subtracts the smaller fraction from the larger one (5/6 minus 1/8) instead of borrowing a whole.
B: The borrow was made in the fraction part, but the whole number was not reduced from 6 to 5.
C: This is the sum of the two lengths; a piece cut off must be subtracted.
D (correct): Rename with denominator 24, borrow 1 from the 6, and subtract: $5 \frac{27}{24} - 2 \frac{20}{24} = 3 \frac{7}{24}$ .

Practice 3 of 35 · easy

Which fraction is equivalent to $\frac{3}{7}$ ?

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Answer: C

Correct answer: 12/28. An equivalent fraction comes from multiplying the numerator and denominator by the same number. Multiplying 3/7 by 4/4 gives 12/28.

A: The numerator was multiplied by 3 but the denominator only by 2, which changes the value.
B: Dividing 6/21 by 3 on top and bottom gives 2/7, not 3/7.
C (correct): Multiplying the top and bottom of 3/7 by 4 gives 12/28, so the value is unchanged.
D: The denominator was multiplied by 4 but the numerator by 5, so this equals more than 3/7.

Practice 4 of 35 · medium

A soldier runs $2 \frac{3}{4}$ miles on Monday and $3 \frac{5}{6}$ miles on Tuesday. What is the total distance for the two days?

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Answer: B

Correct answer: 6 7/12 miles. Add the whole numbers:

2 + 3 = 5

Add the fractions with denominator 12:

9/12 + 10/12 = 19/12

= $1 \frac{7}{12}$ .

Total:

5 + 1 \frac{7}{12} = 6 \frac{7}{12} miles

A: The fraction sum 19/12 is more than 1, but the extra whole was not carried into the whole-number part.
B (correct): Adding wholes gives 5 and adding fractions gives 9/12 + 10/12 = 19/12 = 1 7/12, for 6 7/12 miles in all.
C: This adds the numerators and the denominators (3 + 5 over 4 + 6), which is not allowed with unlike denominators.
D: Only one whole should be carried from 19/12; carrying two adds a mile that is not there.

Practice 5 of 35 · medium

What is $2 \frac{1}{4} \div 1 \frac{1}{2}$ ?

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Answer: D

Correct answer: 1 1/2. Convert to improper fractions, then multiply by the reciprocal of the divisor.

\frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = 1 \frac{1}{2}

A: This flips the first fraction instead of the divisor, giving 4/9 times 3/2.
B: This multiplies 9/4 by 3/2 without flipping the divisor.
C: This divides piece by piece (2 by 1 and 1/4 by 1/2), which does not work for mixed numbers.
D (correct): As improper fractions this is 9/4 times 2/3, which is 18/12 or 1 1/2.

Practice 6 of 35 · easy

A cook measures out $\frac{23}{4}$ cups of broth. What is this amount written as a mixed number?

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Answer: D

Correct answer: 5 3/4 cups. Divide the numerator by the denominator:

23 \div 4 = 5

remainder 3. The quotient is the whole number and the remainder stays over 4, giving $5 \frac{3}{4}$ cups.

A: 4 3/4 equals 19/4; the quotient of 23 by 4 is 5, not 4.
B: The whole part is right, but the remainder is 3, not 1.
C: 6 3/4 equals 27/4, which is 4 quarters more than the measured amount.
D (correct): 23 divided by 4 is 5 with remainder 3, so the amount is 5 3/4 cups.

Practice 7 of 35 · medium

A patrol boat travels $\frac{5}{6}$ mile in $\frac{1}{12}$ hour. What is its speed in miles per hour?

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Answer: B

Correct answer: 10 miles per hour. Rate equals distance divided by time.

\frac{5}{6} \div \frac{1}{12} = \frac{5}{6} \times \frac{12}{1} = 10 miles per hour

A: This uses only 1 divided by 1/12 and leaves out the 5/6 mile distance.
B (correct): Speed is distance divided by time: 5/6 times 12/1 = 10 miles per hour.
C: This multiplies distance by time instead of dividing distance by time.
D: This divides time by distance, which gives hours per mile, not miles per hour.

Practice 8 of 35 · medium

A cabinetmaker has $\frac{3}{4}$ of a sheet of plywood and uses $\frac{2}{3}$ of that piece to build a drawer. What fraction of a full sheet did she use?

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Answer: A

Correct answer: 1/2. The word "of" signals multiplication: $\frac{2}{3} \times \frac{3}{4}$ . Cancel the 3s and simplify to get

\frac{2}{4} = \frac{1}{2}

of a sheet.

A (correct): "Of" means multiply: 2/3 times 3/4 is 6/12, which simplifies to 1/2.
B: This subtracts 2/3 from 3/4, but the problem asks for a fraction of a fraction, which is a product.
C: This divides 3/4 by 2/3; she cannot use more than a full sheet from a 3/4 piece.
D: This is the part of a full sheet that is left over (3/4 minus 1/2), not the part she used.

Practice 9 of 35 · hard

A truck's fuel tank holds 24 gallons and starts a trip $\frac{3}{4}$ full. The trip uses fuel equal to $\frac{5}{8}$ of a full tank, and then the driver adds 9 gallons. What fraction of the tank is full now?

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Answer: A

Correct answer: 1/2. Start:

\frac{3}{4} \times 24 = 18 gallons

The trip uses

\frac{5}{8} \times 24 = 15 gallons

leaving 3 gallons.

After adding 9 gallons there are 12 gallons, and

\frac{12}{24} = \frac{1}{2}

of the tank.

A (correct): The tank goes from 18 gallons to 18 - 15 = 3 gallons, then up to 12 gallons, and 12/24 = 1/2.
B: This takes 5/8 of the fuel in the tank (18 gallons) rather than 5/8 of a full tank.
C: This counts only the 9 added gallons and ignores the 3 gallons already in the tank.
D: This is the fraction left after the trip; the 9 gallons added afterward were left out.

Practice 10 of 35 · medium

A tow rope is $4 \frac{2}{3}$ yards long. How long is it in feet? (1 yard = 3 feet)

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Answer: C

Correct answer: 14 feet. Change the mixed number to an improper fraction and multiply by the conversion factor.

4 \frac{2}{3} \times 3 = \frac{14}{3} \times 3 = 14 feet

A: Only the whole 4 yards were multiplied by 3; the 2/3 yard must be converted too.
B: This divides by 3, but feet are smaller units, so the number of feet must be larger.
C (correct): As an improper fraction the length is 14/3 yards, and 14/3 times 3 = 14 feet.
D: This multiplies by 12, which converts feet to inches, not yards to feet.

Practice 11 of 35 · medium

A generator's fuel tank is $\frac{9}{10}$ full at the start of a shift and $\frac{1}{4}$ full at the end. What fraction of the tank was used during the shift?

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Answer: C

Correct answer: 13/20. Fuel used is the start level minus the end level. With the common denominator 20:

\frac{9}{10} - \frac{1}{4} = \frac{18}{20} - \frac{5}{20} = \frac{13}{20}

A: This subtracts the numerators and the denominators separately (8/6), and more than a full tank cannot have been used.
B: This adds the two fractions instead of finding the difference.
C (correct): With denominator 20 the change is 18/20 - 5/20 = 13/20.
D: This multiplies the fractions; the amount used is the start level minus the end level.

Practice 12 of 35 · easy

A convoy must drive 85 miles to reach a training area. It has covered $\frac{3}{5}$ of the distance. How many miles has it covered?

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Answer: A

Correct answer: 51 miles. "A fraction of" means multiply. Divide by the denominator, then multiply by the numerator:

85 \div 5 = 17 and 17 \times 3 = 51 miles

A (correct): 85 divided by 5 is 17, and 17 times 3 is 51 miles.
B: This is only 1/5 of the trip; it still needs to be multiplied by the numerator 3.
C: This is 2/5 of the trip, the part still left to drive.
D: This is 4/5 of the trip, from multiplying by 4 instead of 3.

Practice 13 of 35 · medium

A mechanic stacks a washer $\frac{5}{16}$ inch thick on a spacer $\frac{3}{8}$ inch thick. How thick is the stack?

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Answer: C

Correct answer: 11/16 inch. Use the common denominator 16:

\frac{3}{8} = \frac{6}{16}

\frac{6}{16} + \frac{5}{16} = \frac{11}{16} inch

A: This adds the numerators and denominators straight across (8/24), which ignores the unlike part sizes.
B: This multiplies the thicknesses; stacking items means adding them.
C (correct): Renaming 3/8 as 6/16 and adding 5/16 gives 11/16 inch.
D: This is the difference between the two thicknesses, not their total.

Practice 14 of 35 · medium

A trainee reads $\frac{1}{4}$ of a manual on Monday. On Tuesday she reads $\frac{1}{3}$ of the pages that were still unread. What fraction of the whole manual is still unread?

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Answer: B

Correct answer: 1/2. After Monday, $\frac{3}{4}$ is unread. Tuesday leaves $\frac{2}{3}$ of that part unread.

\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}

A: This subtracts 1/4 and 1/3 from 1, treating Tuesday's 1/3 as a fraction of the whole manual instead of the unread part.
B (correct): After Monday 3/4 is unread; reading 1/3 of that leaves 2/3 of 3/4, which is 1/2.
C: This is 1/4 + 1/3, which again treats Tuesday's 1/3 as a fraction of the whole manual.
D: This is the part read on Tuesday (1/3 of 3/4), not the part still unread.

Practice 15 of 35 · medium

A warehouse crew has unloaded 45 boxes, which is $\frac{3}{5}$ of a shipment. How many boxes are in the whole shipment?

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Answer: D

Correct answer: 75 boxes. Since 3 fifths make 45 boxes, one fifth is

45 \div 3 = 15

boxes. The whole shipment is

15 \times 5 = 75

boxes, the same as $45 \div \frac{3}{5}$ .

A: This finds 3/5 of 45, but 45 is already 3/5 of the unknown total.
B: This is the 2/5 of the shipment still to unload, not the whole shipment.
C: This is only 1/5 of the shipment; it must be multiplied by 5.
D (correct): If 3/5 of the shipment is 45, then 1/5 is 15 and the whole (5/5) is 75.

Practice 16 of 35 · medium

A recipe calls for $\frac{3}{4}$ cup of sugar per batch. How much sugar is needed for 5 batches?

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Answer: C

Correct answer: 3 3/4 cups. Multiply the amount per batch by the number of batches:

5 \times \frac{3}{4} = \frac{15}{4}

Since

15 \div 4 = 3

remainder 3, this is $3 \frac{3}{4}$ cups.

A: Multiplying both the top and bottom by 5 (15/20) leaves the value unchanged; only the numerator is multiplied by a whole number.
B: This divides 5 by 3/4, which tells how many batches 5 cups would make.
C (correct): 5 times 3/4 is 15/4, and 15 divided by 4 is 3 with remainder 3.
D: This subtracts 3/4 from 5 instead of multiplying for 5 batches.

Practice 17 of 35 · easy

Which fraction is less than $\frac{1}{2}$ ?

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Answer: B

Correct answer: 7/15. Compare each numerator with half of its denominator. Only 7/15 has a numerator (7) smaller than half its denominator (7.5), so it is the only fraction less than 1/2.

A: Half of 9 is 4.5, and 5 is more than that, so 5/9 is greater than 1/2.
B (correct): Half of 15 is 7.5, and 7 is less than 7.5, so 7/15 is below 1/2.
C: Half of 11 is 5.5, and 6 is more than that, so 6/11 is greater than 1/2.
D: Half of 17 is 8.5, and 9 is more than that, so 9/17 is greater than 1/2.

Practice 18 of 35 · easy

What is $\frac{7}{8} - \frac{1}{2}$ ?

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Answer: D

Correct answer: 3/8. Use the common denominator 8:

\frac{1}{2} = \frac{4}{8} then \frac{7}{8} - \frac{4}{8} = \frac{3}{8}

A: This subtracts straight across (6/6); denominators are never subtracted.
B: This renames 1/2 as 2/8, but 1/2 equals 4/8.
C: This adds the fractions (11/8) instead of subtracting.
D (correct): Renaming 1/2 as 4/8 gives 7/8 - 4/8 = 3/8.

Practice 19 of 35 · hard

A carpenter cuts two shelves, $3 \frac{5}{8}$ feet and $4 \frac{3}{4}$ feet long, from a 12-foot board. Each of the two saw cuts turns $\frac{1}{8}$ foot of wood into sawdust. How much of the board is left?

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Answer: D

Correct answer: 3 3/8 feet. Total used:

3 \frac{5}{8} + 4 \frac{6}{8} + \frac{2}{8} = 7 \frac{13}{8} = 8 \frac{5}{8} feet

Then borrow:

12 - 8 \frac{5}{8} = 11 \frac{8}{8} - 8 \frac{5}{8} = 3 \frac{3}{8} feet

A: This subtracts the two shelves but ignores the wood lost in the saw cuts.
B: This counts the sawdust for only one cut; there are two cuts at 1/8 foot each.
C: The fraction was found by borrowing 1 from the 12, but the whole number was not lowered to 11 before subtracting.
D (correct): The shelves and two saw cuts use 3 5/8 + 4 6/8 + 2/8 = 8 5/8 feet, and 12 - 8 5/8 = 3 3/8 feet.

Practice 20 of 35 · easy

What is $\frac{2}{3} \times \frac{9}{10}$ ?

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Answer: A

Correct answer: 3/5. Multiply numerators and denominators, canceling first: 2 and 10 share a factor of 2, and 9 and 3 share a factor of 3.

\frac{2}{3} \times \frac{9}{10} = \frac{1}{1} \times \frac{3}{5} = \frac{3}{5}

A (correct): Multiplying across gives 18/30, which simplifies to 3/5 (or cancel first: 1/1 times 3/5).
B: This adds the numerators and the denominators, which is not how fractions are multiplied.
C: This flips the second fraction, which is the step for division, not multiplication.
D: This is the sum of the fractions (47/30), not the product.

Practice 21 of 35 · medium

A lab technician pours $\frac{1}{3}$ liter of one solution, $\frac{1}{4}$ liter of a second and $\frac{1}{6}$ liter of a third into a beaker. How much liquid is in the beaker?

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Answer: A

Correct answer: 3/4 liter. The smallest common denominator for 3, 4 and 6 is 12.

\frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{9}{12} = \frac{3}{4} liter

A (correct): With denominator 12 the amounts are 4/12 + 3/12 + 2/12 = 9/12, which simplifies to 3/4.
B: This adds the numerators and the denominators, ignoring that the parts are different sizes.
C: This multiplies the three amounts; combining liquids calls for addition.
D: This adds only the first two amounts and leaves out the 1/6 liter.

Practice 22 of 35 · easy

Which decimal is equal to $\frac{7}{8}$ ?

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Answer: C

Correct answer: 0.875. A fraction bar means divide the numerator by the denominator:

7 \div 8 = 0.875

A: This just writes the digits 7 and 8 after the decimal point instead of dividing.
B: This divides 8 by 7, flipping the fraction; 7/8 is less than 1.
C (correct): A fraction means division, and 7 divided by 8 is 0.875.
D: This is the value of 5/8, not 7/8.

Practice 23 of 35 · medium

A roll holds 10 yards of ribbon. Each bow uses $\frac{3}{4}$ yard. How many complete bows can be made from the roll?

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Answer: A

Correct answer: 13 bows. "How many fit" means divide:

10 \div \frac{3}{4} = 10 \times \frac{4}{3} = \frac{40}{3} = 13 \frac{1}{3}

The 1/3 of a bow cannot be finished, so 13 complete bows can be made.

A (correct): 10 divided by 3/4 is 40/3 = 13 1/3, and only 13 bows can be finished.
B: This rounds 13 1/3 up, but there is not enough ribbon left for a 14th bow.
C: This is the exact quotient, but the question asks for complete bows.
D: This multiplies 10 by 3/4; finding how many pieces fit calls for division.

Practice 24 of 35 · medium

A pump fills $\frac{2}{3}$ of a water tank in $\frac{1}{2}$ hour. At the same rate, how long does it take to fill the whole empty tank?

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Answer: B

Correct answer: 45 minutes. Rate:

\frac{2}{3} \div \frac{1}{2} = \frac{4}{3}

tank per hour.

Time for 1 tank:

1 \div \frac{4}{3} = \frac{3}{4} hour and \frac{3}{4} \times 60 = 45 minutes

A: This multiplies 2/3 by 1/2 (1/3 hour), which gives less time than the 30 minutes already spent on part of the tank.
B (correct): The pump fills 2/3 divided by 1/2 = 4/3 tanks per hour, so one tank takes 3/4 hour, or 45 minutes.
C: This treats the rate of 4/3 tanks per hour as if it were 4/3 hours.
D: This assumes the remaining 1/3 of the tank takes another full half hour, but 1/3 is only half of 2/3.

Practice 25 of 35 · hard

A stew recipe that serves 6 people uses $2 \frac{1}{4}$ cups of diced potatoes. How many cups are needed to serve 10 people?

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Answer: A

Correct answer: 3 3/4 cups. Find the amount per person and multiply by 10, which is the same as multiplying by 10/6.

\frac{9}{4} \times \frac{10}{6} = \frac{90}{24} = \frac{15}{4} = 3 \frac{3}{4} cups

A (correct): Scaling by 10/6: 9/4 times 10/6 = 90/24 = 15/4 = 3 3/4 cups.
B: This scales by 6/10 instead of 10/6, which would shrink the recipe for more people.
C: This adds 4 cups because 10 is 4 more than 6, but recipes scale by multiplying, not adding.
D: This multiplies by 10 without dividing by 6, which is enough for 60 people.

Practice 26 of 35 · hard

A soldier's monthly pay is $2,400. She spends $\frac{1}{3}$ of it on rent, then $\frac{1}{4}$ of what remains on food, then saves $\frac{2}{5}$ of what is left after that. How much does she save?

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Answer: A

Correct answer: $480. Work in order, each fraction applying to what remains. Rent:

2400 - 800 = 1600

Food:

\frac{1}{4} \times 1600 = 400

leaving 1,200.

Savings:

\frac{2}{5} \times 1200 = 480

so she saves $480.

A (correct): Rent leaves $1,600, food (1/4 of $1,600) leaves $1,200, and 2/5 of $1,200 is $480.
B: This takes 2/5 of the full $2,400, but the savings fraction applies to what is left after rent and food.
C: This takes 2/5 of the $1,600 left after rent and skips the food step.
D: This is the amount left after saving (3/5 of $1,200), not the amount saved.

Practice 27 of 35 · easy

A training run lasts 40 minutes. What fraction of an hour is this?

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Answer: D

Correct answer: 2/3. An hour has 60 minutes, so 40 minutes is $\frac{40}{60}$ . Divide the top and bottom by 20 to get $\frac{2}{3}$ hour.

A: This treats an hour as 100 minutes (40/100); an hour has 60 minutes.
B: Three-quarters of an hour is 45 minutes, not 40.
C: Five-sixths of an hour is 50 minutes, not 40.
D (correct): 40 out of 60 minutes is 40/60, which simplifies to 2/3 by dividing by 20.

Practice 28 of 35 · hard

In a motor pool, $\frac{2}{5}$ of the vehicles are trucks. The rest are jeeps and vans in the ratio 1:2. If there are 18 vans, how many vehicles are in the motor pool?

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Answer: B

Correct answer: 45 vehicles. Non-trucks are $\frac{3}{5}$ of the total, and vans are $\frac{2}{3}$ of that:

\frac{2}{3} \times \frac{3}{5} = \frac{2}{5}

Total:

18 \div \frac{2}{5} = 18 \times \frac{5}{2} = 45

vehicles (18 trucks, 9 jeeps, 18 vans).

A: This treats the 18 vans as all of the non-truck 3/5, forgetting the jeeps.
B (correct): Vans are 2/3 of the 3/5 that are not trucks, which is 2/5 of all vehicles, and 18 divided by 2/5 is 45.
C: This counts only the jeeps and vans (9 + 18) and leaves out the trucks.
D: This assumes vans are 1/3 of all vehicles, but they make up 2/5 of the total.

Practice 29 of 35 · hard

A convoy covers $13 \frac{1}{2}$ miles in $22 \frac{1}{2}$ minutes. What is its average speed in miles per hour?

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Answer: B

Correct answer: 36 miles per hour. Change the time to hours:

22 \frac{1}{2} \div 60 = \frac{45}{2} \times \frac{1}{60} = \frac{3}{8} hour

\frac{27}{2} \div \frac{3}{8} = \frac{27}{2} \times \frac{8}{3} = 36 miles per hour

A: This is miles per minute (13 1/2 divided by 22 1/2); the minutes must first be changed to hours.
B (correct): 22 1/2 minutes is 3/8 hour, and 27/2 divided by 3/8 is 27/2 times 8/3 = 36.
C: This treats 22 1/2 minutes as half an hour, but half an hour is 30 minutes.
D: This multiplies the distance by the time in hours instead of dividing.

Practice 30 of 35 · easy

In a company's vehicle fleet, the ratio of diesel trucks to gasoline trucks is 3:4. What fraction of the trucks are diesel?

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Answer: A

Correct answer: 3/7. Add the ratio parts to get the whole:

3 + 4 = 7

Diesel trucks are 3 of those 7 parts, so they make up $\frac{3}{7}$ of the trucks.

A (correct): The ratio has 3 + 4 = 7 parts in all, and 3 of them are diesel.
B: This compares diesel to gasoline trucks; a fraction of the total must use all 7 parts.
C: This is the fraction of trucks that run on gasoline.
D: This is the gasoline-to-diesel ratio written as a fraction; a part of a group cannot be more than the whole group.

Practice 31 of 35 · hard

A supply depot issues $\frac{3}{7}$ of its cases of rations, then issues $\frac{1}{2}$ of the cases that remain. 120 cases are left. How many cases did the depot have at the start?

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Answer: B

Correct answer: 420 cases. After the first issue $\frac{4}{7}$ remains; after the second, half of that, $\frac{2}{7}$ , remains.

120 \div \frac{2}{7} = 120 \times \frac{7}{2} = 420 cases

A: This subtracts 3/7 and 1/2 from 1 as if both were fractions of the starting amount, giving 1/14.
B (correct): The 120 cases are 1/2 of 4/7, or 2/7, of the starting amount, and 120 divided by 2/7 is 420.
C: This undoes only the second issue; 240 is the number of cases after the first issue.
D: This divides 120 by 3/7, the fraction issued first, instead of by the fraction left.

Practice 32 of 35 · hard

A room measures $12 \frac{1}{2}$ feet by $9 \frac{3}{5}$ feet. Carpet is sold by the square yard, and 9 square feet make 1 square yard. How many square yards of carpet cover the floor exactly?

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Answer: B

Correct answer: 13 1/3 square yards. Convert and multiply:

\frac{25}{2} \times \frac{48}{5} = \frac{1200}{10} = 120 square feet

Then

120 \div 9 = \frac{40}{3} = 13 \frac{1}{3} square yards

A: This multiplies the mixed numbers piece by piece (12 times 9 plus 1/2 times 3/5) instead of converting to improper fractions first.
B (correct): The area is 25/2 times 48/5 = 120 square feet, and 120 divided by 9 is 13 1/3 square yards.
C: This divides the 120 square feet by 3; 3 feet make a yard of length, but 9 square feet make a square yard.
D: This multiplies by 9 instead of dividing; square yards are larger units, so the count must be smaller.

Practice 33 of 35 · medium

A motor pool has 156 vehicles. Of these, $\frac{7}{12}$ are trucks and the rest are cars. How many cars are there?

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Answer: C

Correct answer: 65 cars. Cars are

1 - \frac{7}{12} = \frac{5}{12}

of the vehicles. Then

156 \div 12 = 13 and 13 \times 5 = 65

cars.

A: This is the number of trucks (7/12 of 156), not the number of cars.
B: This is only 1/12 of the vehicles; cars make up 5 twelfths.
C (correct): One twelfth of 156 is 13; cars are 5/12 of the total, and 13 times 5 is 65.
D: This subtracts just one twelfth (13) from 156 instead of the 7 twelfths that are trucks.

Practice 34 of 35 · hard

One pipe fills $\frac{1}{6}$ of a tank each hour, and a second pipe fills $\frac{1}{4}$ of the tank each hour. Both pipes run for 2 hours into the empty tank. What fraction of the tank is still empty?

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Answer: D

Correct answer: 1/6. Combined rate:

\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}

tank per hour.

In 2 hours:

2 \times \frac{5}{12} = \frac{5}{6}

filled, so

1 - \frac{5}{6} = \frac{1}{6}

is still empty.

A: This adds the hourly rates straight across (2/10) before doubling, which does not give the true combined rate.
B: This uses the work of only 1 hour; the pipes run for 2 hours.
C: This is the fraction of the tank that is filled, not the part still empty.
D (correct): Together the pipes fill 5/12 per hour, so 2 hours fill 10/12 = 5/6, and 1/6 is still empty.

Practice 35 of 35 · medium

Which fraction lies between $\frac{1}{3}$ and $\frac{1}{2}$ ?

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Answer: D

Correct answer: 2/5. Use 1/2 as a benchmark: 2/5 is below it because 2 is less than half of 5. Then compare with 1/3 by cross-multiplying:

2 \times 3 = 6 and 1 \times 5 = 5

so 2/5 is greater than 1/3.

A: Cross-multiplying with 1/3 gives 9 versus 10, so 3/10 is less than 1/3.
B: Its numerator 5 is more than half of 9, so 5/9 is greater than 1/2.
C: With equal numerators, the larger denominator gives the smaller fraction, so 1/4 is less than 1/3.
D (correct): Its numerator is less than half its denominator, so it is below 1/2, and cross-multiplying with 1/3 (6 versus 5) shows it is above 1/3.

About this test

After each response you get a written explanation showing the setup and the arithmetic, not just the correct letter. Read the explanation even when you got the item right, since it may show a faster route than the one you used.

Use your score as a map. If you missed questions on dividing fractions, go back to that lesson section; if mixed measurements tripped you up, revisit the worked example on that topic. When you want more repetitions, 35 extra practice questions for this lesson are available. Your accuracy here is a study signal only and is not an ASVAB or AFQT score.

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