Testing a proportion: cross multiplication, tables and graphs
Mathematics Prep 1 Term 1, Unit 1, Lesson 1
GoalTest whether two ratios form a proportion using cross products, a table or a graph.
New words
cross multiplication · multiplying the extremes and multiplying the meansconstant ratio · a ratio that stays the same in every rowproportional · two quantities whose ratio never changesorigin · the point (0, 0) where the axes meet
Explanation
To test a : b and c : d, multiply the extremes a × d and the means b × c. If the two products are equal, the ratios form a proportion. Why does it work? Write a/b = c/d and multiply both sides by b × d: this gives a × d = c × b. You can also test with a table: if the ratio of the two quantities is the same in every row, they are proportional. On a graph, proportional quantities lie on a straight line that passes through the origin (0, 0). A straight line that does not pass through the origin is not a proportion.
ƒFormula
a : b = c : d means a × d = b × c
Cross products are equal.
Worked examples
Test 3 : 5 and 12 : 20. Extremes: 3 × 20 = 60. Means: 5 × 12 = 60. The products are equal, so 3 : 5 = 12 : 20 is a proportion.
Test 4 : 7 and 9 : 16. Extremes: 4 × 16 = 64. Means: 7 × 9 = 63. 64 is not 63, so these ratios are not equal and there is no proportion.
Mass (kg)
Price (LE)
Price ÷ mass
2
24
12
3
36
12
5
60
12
A table of oranges: 2 kg cost 24 LE, 3 kg cost 36 LE, 5 kg cost 60 LE. Price ÷ mass is 12 in every row, so the price is proportional to the mass. In another table the heights of a plant are 4 cm after 1 week, 8 cm after 2 weeks and 11 cm after 3 weeks. The ratios 4, 4 and about 3.7 change, so this is not a proportion.
Class activity
Shop prices. With a grown-up, look at a price list for 1, 2 and 3 items of the same kind (or make one up). Write a table of items and price. Is the price proportional to the number of items? Check the ratio in every row. Then add a delivery fee of 5 LE: what changes?
Practice
1
Is 6 : 9 = 8 : 12 a proportion? Use cross products.
Yes. 6 × 12 = 72 and 9 × 8 = 72.
2
Is 5 : 6 = 7 : 9 a proportion?
No. 5 × 9 = 45 but 6 × 7 = 42.
3
Time (h): 1, 2, 4 and distance (km): 70, 140, 280. Are they proportional?
Yes. 70 ÷ 1 = 140 ÷ 2 = 280 ÷ 4 = 70 km per hour.
4
Notebooks: 3, 6, 9 and price (LE): 15, 30, 42. Are they proportional?
No. 15 ÷ 3 = 5 and 30 ÷ 6 = 5, but 42 ÷ 9 is not 5.
5
Youssef checks 2 : 3 and 6 : 9 by 2 × 3 and 3 × 6. What did he do wrong? What is the right test?
He multiplied the first terms of each ratio instead of the extremes and means. The test is 2 × 9 = 18 and 3 × 6 = 18: equal, so it is a proportion.
