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Solving a proportion for the missing term in word problems
Mathematics Prep 1 Term 1, Unit 1, Lesson 1
GoalFind the missing term of a proportion by cross multiplication and use it in word problems.
New words
unknown · the missing number, often written xrate · how much of one quantity for one of anothersame order · the same quantity on top (or first) in both ratioscheck · test the answer in the original proportion
Explanation
To find a missing term, cross multiply and divide. For 3 : 5 = 12 : x, the cross products give 3 × x = 5 × 12, so x = 60 ÷ 3 = 20. You can sometimes simplify first, which makes the numbers smaller. In a word problem, write the two quantities in the same order in both ratios. If 4 books cost 52 LE and you want the price of 7 books, write books : price = 4 : 52 = 7 : x. Then check: 20 gives 3 × 20 = 60 = 5 × 12.
★Remember
Cross multiply, then divide by the number next to x. Keep the same order in both ratios.
Worked examples
Solve 3 : 5 = 12 : x. Cross products: 3 × x = 5 × 12 = 60. So x = 60 ÷ 3 = 20. Check: 3 × 20 = 60 and 5 × 12 = 60.
Solve x/8 = 15/24. Simplify first: 15/24 = 5/8, so x/8 = 5/8 and x = 5. With cross products: 24 × x = 8 × 15 = 120 and x = 120 ÷ 24 = 5. Both ways agree.
4 books cost 52 LE. What do 7 books cost at the same rate? books : price = 4 : 52 = 7 : x. Cross products: 4 × x = 52 × 7 = 364, so x = 364 ÷ 4 = 91. Seven books cost 91 LE. Check: one book costs 13 LE and 13 × 7 = 91.
Class activity
Recipe scaling. A recipe for 4 people uses 6 cups of rice. With a grown-up, write a proportion to find the rice for 10 people, solve it, and check by finding the rice for one person first.
Practice
1
Solve 6 : 9 = 10 : x.
x = 9 × 10 ÷ 6 = 15.
2
Solve x : 12 = 7 : 4.
4 × x = 12 × 7 = 84, so x = 21.
3
5 notebooks cost 35 LE. How much do 8 notebooks cost?
5 : 35 = 8 : x, so x = 35 × 8 ÷ 5 = 56. They cost 56 LE.
4
A train travels 150 km in 2 hours. At the same speed, how long does it take to travel 375 km?
150 : 2 = 375 : x, so x = 2 × 375 ÷ 150 = 5. It takes 5 hours.
5
Seif solves 4 : 6 = x : 15 and writes x = 4 × 6 ÷ 15. What is wrong? Find x.
He multiplied the two numbers on the same side. The cross products are 4 × 15 = 6 × x, so x = 60 ÷ 6 = 10.