Study Companion
Topic 03 | Numerical reasoning

Quantitative word problems

Selective prep: ratio, proportion and rate word problems solvable by primary-school arithmetic — no formal algebra needed.

20-30 min Printable practice Answer key Challenge included
How to use this page

Read the explanation, work through the examples, then complete the core practice before printing.

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What you will learn

Worked example 0 Unitary method

Question: 4 pens cost $6. How much do 10 pens cost?   (a) $12 (b) $14 (c) $15 (d) $16

Step 1. Cost of 1 pen =6÷4=1.50= 6 \div 4 = 1.50, i.e. $1.50.

Step 2. Cost of 10 pens =10×1.50=15= 10 \times 1.50 = 15, i.e. $15.

Step 3. Estimate check: 10 pens is 2.52.5 times the first batch, so cost should be about 2.5×6=152.5 \times 6 = 15 dollars. Matches. Answer: (c).

1. Unitary method

If you know the amount for “many”, first find the amount for “one”.

Worked example 1 Scaling up a recipe

A recipe for 6 people uses 300 g of rice. How much rice for 10 people?

Rice per person =300÷6=50= 300 \div 6 = 50 g. For 10 people =10×50=500= 10 \times 50 = \mathbf{500} g.

2. Sharing in a ratio

Split a total by first adding the ratio parts, then dividing the total by that sum.

Worked example 2 Share $60 in ratio 2 : 3

Total parts =2+3=5= 2 + 3 = 5. One part =60÷5=12= 60 \div 5 = 12 dollars. So the shares are 2×12=242 \times 12 = 24 dollars and 3×12=363 \times 12 = 36 dollars.

Check: 24+36=6024 + 36 = 60 and 24:36=2:324 : 36 = 2 : 3.

3. Rate problems

A rate is an amount per unit time (or per unit item). Write the per-unit value explicitly.

Worked example 3 Filling a tank

A tap fills a tank in 20 minutes. A second tap fills the same tank in 30 minutes. Running together, how long to fill it?

Step 1. Per-minute rates. Tap 1 fills 120\tfrac{1}{20} of the tank per minute. Tap 2 fills 130\tfrac{1}{30}.

Step 2. Combined rate = 120+130=360+260=560=112\tfrac{1}{20} + \tfrac{1}{30} = \tfrac{3}{60} + \tfrac{2}{60} = \tfrac{5}{60} = \tfrac{1}{12} of the tank per minute.

Step 3. Time to fill one whole tank = 1÷112=121 \div \tfrac{1}{12} = \mathbf{12} minutes.

4. Percentage shortcuts

Convert the percent to a fraction you can handle mentally.

Worked example 4 Discount and add-back

A bag originally costs $40. It is discounted by 25%25\%, then a $5 voucher is applied. What is the final price?

25%25\% of $40 is 14×40=10\tfrac{1}{4} \times 40 = 10 dollars. After discount: 40−10=3040 - 10 = 30 dollars. After voucher: 30−5=2530 - 5 = \mathbf{25} dollars.

Trap: don’t apply the voucher first — the problem specifies the order.

Drill pack

10 minutes. Write the per-unit value or per-part value before you start each question. Pencil, no calculator.

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Fluency

Timed drill

    1. 5 apples cost $4. How much for 15 apples? (a) $10 (b) $12 (c) $15 (d) $20
    2. A recipe for 4 people uses 200 g of flour. How much for 6 people? (a) 250 g (b) 280 g (c) 300 g (d) 320 g
    3. Share $80 between two friends in the ratio 3:53 : 5. How much does the second friend get? (a) $30 (b) $40 (c) $45 (d) $50
    4. A car travels 120 km in 2 hours. How far in 5 hours at the same speed? (a) 240 km (b) 260 km (c) 300 km (d) 360 km
    5. 20%20\% of a class of 30 students walk to school. How many walk? (a) 4 (b) 5 (c) 6 (d) 8
    6. A shirt marked $60 is discounted by 25%25\%. What is the sale price? (a) $35 (b) $40 (c) $45 (d) $50
    7. Three taps each fill a tank in 30 minutes. Running together, how long? (a) 5 min (b) 10 min (c) 15 min (d) 20 min
    8. 8 workers paint a fence in 6 days. How many days would 12 workers take (same pace)? (a) 3 (b) 4 (c) 5 (d) 9
    9. Share 45 lollies between three children in the ratio 1:2:21 : 2 : 2. How many does the smallest-share child get? (a) 5 (b) 8 (c) 9 (d) 15
    10. A train travels at 80 km/h. How far in 45 minutes? (a) 40 km (b) 50 km (c) 55 km (d) 60 km
    11. A dripping tap loses 2 litres per hour. How much in a full day? (a) 24 L (b) 36 L (c) 48 L (d) 60 L
    12. A $50 item has GST of 10%10\% added. What is the new price? (a) $52 (b) $55 (c) $57 (d) $60
    13. The ratio of boys to girls in a class is 2:32 : 3. If there are 10 boys, how many students are there in total? (a) 15 (b) 20 (c) 25 (d) 30
    14. 6 books fit in a box. How many boxes for 100 books? (a) 15 (b) 16 (c) 17 (d) 18
    15. A recipe says “1 cup flour to 3 cups water”. For 2 cups of flour, how many cups of water? (a) 2 (b) 4 (c) 5 (d) 6
    16. Alice earns $12 an hour. How much for a 7.5 hour shift? (a) $84 (b) $88 (c) $90 (d) $96
    17. Half a pizza has 8 slices. How many slices in 3 whole pizzas? (a) 24 (b) 36 (c) 40 (d) 48
    18. A class of 40 has 60%60\% girls. How many boys? (a) 12 (b) 16 (c) 18 (d) 24
    19. A stamp costs 35 cents. Ten stamps cost? (a) $3.50 (b) $3.65 (c) $3.75 (d) $4.00
    20. Three friends share a bill in the ratio 1:2:31 : 2 : 3. If the total is $60, how much does the middle-share friend pay? (a) $10 (b) $15 (c) $20 (d) $30

Challenge

Two-step problems: still solvable without algebra if you think in unit chunks.

Reasoning

Harder patterns

    1. A tap fills a tank in 30 min; a drain empties it in 20 min. Both open on a full tank. How long until empty? (a) 30 min (b) 40 min (c) 50 min (d) 60 min
    2. 4 workers build a wall in 9 days. How long for 6 workers at the same pace? (a) 5 days (b) 6 days (c) 7 days (d) 8 days
    3. A shopkeeper buys a toy for $20 and sells it for $25. What is the percentage profit? (a) 15%15\% (b) 20%20\% (c) 25%25\% (d) 30%30\%
    4. The ratio of cats to dogs in a shelter is 2:52 : 5. There are 21 more dogs than cats. How many cats are there? (a) 12 (b) 14 (c) 16 (d) 18
    5. A jug holds 1.5 L of juice. Sam fills cups of 250 mL. How many full cups? (a) 5 (b) 6 (c) 7 (d) 8
    6. A $120 jacket is discounted by 25%25\%, then a further 20%20\% off the discounted price. Final price? (a) $66 (b) $72 (c) $80 (d) $84
    7. Two cars leave the same town in opposite directions at 40 km/h and 60 km/h. After how many hours are they 250 km apart? (a) 2 (b) 2.5 (c) 3 (d) 4
    8. In a bag, the ratio of red to blue marbles is 3:43 : 4. There are 28 marbles total. If 7 red marbles are added, what is the new ratio? (a) 5:45 : 4 (b) 4:34 : 3 (c) 3:23 : 2 (d) 19:1619 : 16
Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Each answer shows the one-unit value or one-part value first, then the scaling.

Timed drill

    1. (b) $12. One apple costs 4÷5=804 \div 5 = 80 cents. Fifteen apples: 15×0.80=1215 \times 0.80 = 12 dollars.
    2. (c) 300 g. One person =200÷4=50= 200 \div 4 = 50 g. Six people =6×50=300= 6 \times 50 = 300 g.
    3. (d) $50. Total parts =3+5=8= 3 + 5 = 8. One part =80÷8=10= 80 \div 8 = 10 dollars. Second friend =5×10=50= 5 \times 10 = 50 dollars.
    4. (c) 300 km. Speed =120÷2=60= 120 \div 2 = 60 km/h. Distance =5×60=300= 5 \times 60 = 300 km.
    5. (c) 6. 20%=1520\% = \tfrac{1}{5}. 15×30=6\tfrac{1}{5} \times 30 = 6.
    6. (c) $45. 25%=1425\% = \tfrac{1}{4}. Discount =15= 15 dollars. Sale price =60−15=45= 60 - 15 = 45 dollars.
    7. (b) 10 min. Each tap fills 130\tfrac{1}{30} per min; three taps fill 330=110\tfrac{3}{30} = \tfrac{1}{10} per min. Time =10= 10 min.
    8. (b) 4. Worker-days =8×6=48= 8 \times 6 = 48. 48÷12=448 \div 12 = 4 days.
    9. (c) 9. Total parts =1+2+2=5= 1 + 2 + 2 = 5. One part =45÷5=9= 45 \div 5 = 9. Smallest share =1×9=9= 1 \times 9 = 9.
    10. (d) 60 km. 4545 min =34= \tfrac{3}{4} hour. Distance =80×34=60= 80 \times \tfrac{3}{4} = 60 km.
    11. (c) 48 L. 2424 hours ×2\times 2 L/hour =48= 48 L.
    12. (b) $55. 10%10\% of 5050 is 55 dollars. New price =55= 55 dollars.
    13. (c) 25. Ratio 2:32 : 3, so 22 parts =10= 10 boys, 11 part =5= 5. Total 55 parts =25= 25.
    14. (c) 17. 100÷6=16100 \div 6 = 16 remainder 44. The last 44 books still need a box, so 1717 boxes.
    15. (d) 6. Flour doubles, so water doubles too: 3×2=63 \times 2 = 6 cups.
    16. (c) $90. 12×7.5=9012 \times 7.5 = 90 dollars.
    17. (d) 48. One pizza =16= 16 slices. 33 pizzas =48= 48 slices.
    18. (b) 16. If 60%60\% are girls, 40%40\% are boys. 40%40\% of 40=1640 = 16.
    19. (a) $3.50. 10×0.35=3.5010 \times 0.35 = 3.50 dollars.
    20. (c) $20. Total parts =1+2+3=6= 1 + 2 + 3 = 6. One part =60÷6=10= 60 \div 6 = 10 dollars. Middle share =2×10=20= 2 \times 10 = 20 dollars.

Challenge

    1. (d) 60 min. Net drain rate =120−130=360−260=160= \tfrac{1}{20} - \tfrac{1}{30} = \tfrac{3}{60} - \tfrac{2}{60} = \tfrac{1}{60} per minute. So 6060 minutes to empty the full tank.
    2. (b) 6 days. Worker-days =4×9=36= 4 \times 9 = 36. 36÷6=636 \div 6 = 6 days.
    3. (c) 25%25\%. Profit =5= 5 dollars. Percent profit =5/20=14=25%= 5 / 20 = \tfrac{1}{4} = 25\%.
    4. (b) 14. Ratio 2:52 : 5, so difference == 33 parts =21= 21, giving 11 part =7= 7. Cats =2×7=14= 2 \times 7 = 14.
    5. (b) 6. 1.51.5 L =1500= 1500 mL. 1500÷250=61500 \div 250 = 6 full cups.
    6. (b) $72. After first discount: 120×0.75=90120 \times 0.75 = 90 dollars. After second: 90×0.80=7290 \times 0.80 = 72 dollars.
    7. (b) 2.5 hr. Opposite directions, so combined speed =40+60=100= 40 + 60 = 100 km/h. Time =250÷100=2.5= 250 \div 100 = 2.5 hours.
    8. (d) 19:1619 : 16. Ratio 3:43 : 4 of 2828 marbles gives 1212 red, 1616 blue. After adding 77 red: 1919 red, 1616 blue.

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