In proportion
Same simplest form means same taste, same look: a : b :: c : d.
Kesang's lemonade
Kesang makes 6 glasses of lemonade with 10 spoons of sugar. Her friend makes 9 glasses with 15 spoons of sugar.
Do the two lemonades taste equally sweet?
What this lesson covers
The idea
Two ratios are proportional, or in proportion, when their terms change by the same factor, that is, when their simplest forms are the same; this is written a : b :: c : d.
Kesang's lemonade
Kesang makes 6 glasses of lemonade with 10 spoons of sugar. Her friend makes 9 glasses with 15 spoons of sugar.
Do the two lemonades taste equally sweet?
- Yes, equally sweet
- The friend's is sweeter
- Kesang's is sweeter
Which recipes taste the same?
Tap two recipes. Each ratio is reduced by its HCF. If the two simplest forms are the same, the recipes taste the same.
Same simplest form
Two ratios are proportional, or in proportion, when their terms change by the same factor, that is, when their simplest forms are the same; this is written a : b :: c : d.
6 : 10 and 9 : 15 both reduce to 3 : 5. So 6 : 10 :: 9 : 15.
The book's pictures: A, C and D are 60 : 40 :: 30 : 20 :: 90 : 60, all 3 : 2. Picture B is 2 : 1 and E is 1 : 1, so they are not in proportion with A.
Notes
Two ratios are proportional, or in proportion, when their terms change by the same factor, that is, when their simplest forms are the same; this is written a : b :: c : d.
Check yourself
Which ratio is in proportion with 3 : 4?
Picture A is 60 : 40 and picture B is 40 : 20 (width : height). Which sentence is right?
How many of 4 : 6, 9 : 12, 6 : 9, 10 : 15 and 8 : 10 are in proportion with 2 : 3?
Answer: 3
4 : 6, 6 : 9 and 10 : 15 all reduce to 2 : 3. 9 : 12 is 3 : 4 and 8 : 10 is 4 : 5. So the answer is 3.
By what factor are both terms of 6 : 10 multiplied to get 9 : 15?
Answer: 3/2
6 × 3/2 = 9 and 10 × 3/2 = 15. The same factor 3/2 works on both terms, so 6 : 10 :: 9 : 15.
- 4 : 3. The terms are in the wrong order. 4 : 3 is already in simplest form and is not 3 : 4.
- 6 : 9. 6 : 9 reduces to 2 : 3, not 3 : 4.
- 9 : 12 — correct. Yes! 9 : 12 reduces by 3 to 3 : 4.
- 12 : 20. 12 : 20 reduces by 4 to 3 : 5, not 3 : 4.
- They are in proportion: 20 mm was taken off both sides. Taking off the same amount does not decide proportion. Compare the simplest forms.
- They are not in proportion: 3 : 2 and 2 : 1 are different — correct. Yes! 60 : 40 reduces to 3 : 2 but 40 : 20 reduces to 2 : 1.
- They are in proportion: both are rectangles. Both are rectangles, but the simplest forms 3 : 2 and 2 : 1 are different.