Proportional Reasoning-2 · Lesson 1 of 7
Proportional Relationships — A Quick Recap
“Review how ratios describe proportional relationships and how cross-multiplication helps us test whether two ratios are equivalent.”
- Explain what it means for two related quantities to be proportional.
- Represent proportional relationships using ratio notation.
- Recognise equivalent ratios by scaling both terms by the same factor.
- Use cross-multiplication to check whether two ratios are proportional.
- Find a missing value in a simple proportion.
What Makes a Relationship Proportional?
Ratios are useful when two quantities are linked and we want to compare them. A relationship is proportional when the quantities keep the same relative size as they change. If one quantity is multiplied or divided by a factor, the corresponding quantity must change by the same factor.
Think about a mixture made with 2 cups of rice for every 1 cup of urad dal. The ratio is 2 : 1. If we make three times as much mixture, we use 6 cups of rice and 3 cups of dal. Both quantities were multiplied by 3, so the taste-producing proportion has stayed the same.
A relationship in which corresponding quantities change by the same scale factor, so their ratio stays equivalent.
Problem
One mixture uses 6 cups of rice and 3 cups of dal. Another uses 4 cups of rice and 2 cups of dal. Are the two ratios proportional?
- 1.Write the ratios as 6 : 3 and 4 : 2.
- 2.Simplify 6 : 3 by dividing both terms by 3 to get 2 : 1.
- 3.Simplify 4 : 2 by dividing both terms by 2 to get 2 : 1.
- 4.Both reduce to the same ratio, so the two mixtures are proportional.
Checking Proportions by Cross-Multiplication
Simplifying ratios is often enough, but cross-multiplication gives a quick test that works even when the numbers do not simplify immediately. The idea is that two fractions are equal exactly when their cross-products are equal.
For 6 : 3 and 4 : 2, the cross-products are 6 × 2 = 12 and 3 × 4 = 12. Since they are equal, the two ratios are proportional. If the cross-products are different, the ratios are not proportional.
Problem
Are 3 : 5 and 12 : 20 proportional?
- 1.Cross-multiply: 3 × 20 = 60.
- 2.Cross-multiply the other way: 5 × 12 = 60.
- 3.The products are equal, so 3 : 5 :: 12 : 20.
Problem
Find x if 7 : 9 :: 21 : x.
- 1.Write the cross-product equation: 7 × x = 9 × 21.
- 2.Calculate the right side: 9 × 21 = 189.
- 3.Divide both sides by 7: x = 189 ÷ 7 = 27.
- 4.Check: 7 : 9 and 21 : 27 both simplify to the same ratio.
When scaling a ratio, both terms must be multiplied or divided by the same non-zero factor. Changing only one term changes the ratio itself.
Quiz
Which pair of ratios is proportional?
If 4 : 7 :: 12 : x, what is x?
Why are 6 : 3 and 4 : 2 proportional?
For a : b and c : d to be proportional, which equality must hold?
A drink uses syrup and water in the ratio 1 : 4. Which mixture keeps the same proportion?
Practice Problems
- Decide whether 8 : 12 and 10 : 15 are proportional. Show the cross-products.
- Find x if 5 : 8 :: 20 : x.
- A recipe uses flour and sugar in the ratio 3 : 2. Write the ratio after both quantities are multiplied by 4.
- Explain why 4 : 6 and 10 : 14 are not proportional.
- A paint mixture uses 2 parts yellow for every 5 parts blue. If 15 parts of blue are used, how many parts of yellow are needed?
- Create two different ratios that are both proportional to 7 : 3 and explain the scale factor used for each.
Key Takeaways
A proportional relationship keeps the same relative comparison between quantities. Equivalent ratios are formed by multiplying or dividing every term by the same non-zero factor. Two ratios a : b and c : d are proportional when a × d = b × c. Cross-multiplication can test a proportion or help find a missing value. A proportion describes a relationship, not just a coincidence between numbers.
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Ratios in Maps