Write down what does not change
Every word problem here has one quantity that stays the same while everything around it moves. Find it and the problem becomes one equation.
Salt water: the amount of salt
Speed: the distance
Work: the part of the job done
Counting: the choices at each step
Name the unknown first. Write the thing that stays the same on both sides of an equals sign. Solve it. Then check the unit: grams, kilometres, minutes, hours or ways.
Two problems, one equation each
300 g of salt water at 8% is poured into 200 g at 13%. How strong is the mixture?
300 × 8 ÷ 100 = 24 g of salt
200 × 13 ÷ 100 = 26 g of salt
(24 + 26) ÷ 500 × 100 = 10%
Two people start 24 km apart and walk towards each other at 5 km/h and 7 km/h. How long until they meet?
The gap closes at 5 + 7 = 12 km/h
24 ÷ 12 = 2 hours, so 120 minutes
In the first, the amount of salt is conserved. In the second, the two distances walked must add up to the original 24 km gap. The remaining gap shrinks. Speeds add, times do not.
Write the model before calculating
A rental costs 12 credits plus 4 per hour. A total of 36 credits means 12 + 4x = 36, where x is hours. Subtract the fixed charge before dividing by the hourly rate.
x = (36 − 12) ÷ 4 = 6
Eight tickets cost 41 credits. Standard tickets cost 7 and reduced tickets cost 4. Let x and y be their counts. Use both the count and the cost, not just one total.
x + y = 8; 7x + 4y = 41
3x = 41 − 4 × 8 = 9; x = 3; y = 5
Subtract four times the count equation from the cost equation. Three standard tickets and five reduced tickets satisfy both original totals.
A well-mixed 200 g solution contains 10% salt, or 20 g. Remove 50 g: this removes 5 g of salt. Replace it with 50 g of pure water. There are now 15 g of salt in 200 g, giving 7.5%.
10% × (150 ÷ 200) = 7.5%
Check: repeat the same 50 g removal and water replacement after mixing. What is the new salt percentage?
The second removal keeps three quarters of the remaining salt. The result is 5.625%, not 5%: each removal uses the current concentration.
7.5% × (150 ÷ 200) = 5.625%
Set up one more
Work with rates, not times. Take the tank as 12 parts. The first pipe puts in 2 parts an hour and the second 1.
Try it
One pipe fills a tank in 6 hours. Another fills it in 12 hours. How long do the two take together?
- 4 hours
- 9 hours
Show the answer
4 hours
2 + 1 = 3 parts an hour, and 12 ÷ 3 = 4 hours. Rates add, so the pair is faster than either pipe alone.
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Practice Exam word problems