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Equivalent ratios

Ratio, rates and money · level 3, lesson 2 of 6

A question to start

Two friends make lemonade.

  • Ana uses 2 cups of lemon juice and 3 cups of water.
  • Ben uses 3 cups of lemon juice and 5 cups of water.

Ben used more lemon juice. So is Ben's lemonade more lemony? But Ben also used more water. Hmm.

Which jug tastes stronger of lemon, or do they taste exactly the same? You cannot just look at the lemon, and you cannot just look at the water. You have to look at how they are mixed. By the end of this lesson you will have two different ways to decide, and they will agree.

The idea

Making more of the same mix

Ana's recipe is lemon : water \(= 2 : 3\). Suppose she wants a bigger jug that tastes exactly the same. She can simply make the recipe twice and pour both into one jug. That is \(2 + 2 = 4\) cups of lemon and \(3 + 3 = 6\) cups of water.

1 batch 2 : 3 2 batches 4 : 6

Every sip of the big jug is the same as a sip of the small one, because the big jug is just several small jugs poured together. So \(2 : 3\) and \(4 : 6\) describe the same mix. Three batches give \(6 : 9\), ten batches give \(20 : 30\).

Ratios that describe the same mix are called equivalent ratios:

\[2 : 3 = 4 : 6 = 6 : 9 = 20 : 30.\]

The rule, and why it works

Multiplying both parts of a ratio by the same number gives an equivalent ratio.

Multiplying by 3 is the same as making 3 batches. Every batch has the same mix, so the whole jug has the same mix. That is the whole reason.

It works backwards too. If you have a big jug of \(20 : 30\), you could pour it into 10 equal small jugs, and each small jug would have \(2 : 3\). So:

Dividing both parts by the same number gives an equivalent ratio.

The important words are both parts and the same number. If you multiply one part by 3 and the other by 2, the mix changes.

Adding does not work

What if Ana adds 1 cup of lemon and 1 cup of water to her \(2 : 3\) jug? Now she has \(3 : 4\). Is that the same taste?

No. In \(2 : 3\), lemon is \(\tfrac25\) of the drink. In \(3 : 4\), lemon is \(\tfrac37\) of the drink. Since \(\tfrac25 = \tfrac{14}{35}\) and \(\tfrac37 = \tfrac{15}{35}\), the new drink is a little more lemony. Adding the same amount to both parts changes the mix. Ratios are about multiplying, not adding. This is the big idea of the whole topic.

A ratio table

A neat way to list equivalent ratios is a ratio table. Each column is the same mix.

Lemon (cups) 2 4 6 8 10 1
Water (cups) 3 6 9 12 15 1.5

Look at the last column. Dividing both parts of \(2 : 3\) by 2 gives \(1 : 1.5\). That is "1 cup of lemon for every one and a half cups of water". The parts of a ratio do not have to be whole numbers.

Simplest form

Among all the equivalent ratios, one is the easiest to read: the one with the smallest whole numbers. It is called the simplest form. A ratio is in simplest form when its parts are whole numbers and no number bigger than 1 divides all of them.

To simplify, divide both parts by a number that goes into both. Keep going until you cannot:

\[18 : 24 \;\to\; 9 : 12 \;\to\; 3 : 4.\]

(First we divided by 2, then by 3.) The quick way is to divide by the biggest number that goes into both, called the greatest common factor (or greatest common divisor). For 18 and 24 it is 6, and \(18 \div 6 = 3\), \(24 \div 6 = 4\), giving \(3 : 4\) in one step. You can learn all about it in Number theory, level 4.

Ratios with decimals or fractions

If a ratio has decimals, multiply both parts to get whole numbers, then simplify:

  • \(1.5 : 2\): multiply both by 2 to get \(3 : 4\).
  • \(0.4 : 1.2\): multiply both by 10 to get \(4 : 12\), then divide by 4 to get \(1 : 3\).

If it has fractions, multiply by a number that clears all the bottoms:

  • \(\tfrac12 : \tfrac13\): multiply both by 6. \(\tfrac12 \times 6 = 3\) and \(\tfrac13 \times 6 = 2\), so \(3 : 2\).

Comparing two mixes

Back to the lemonade. Ana is \(2 : 3\), Ben is \(3 : 5\). Two good ways to compare:

Way 1: make one part the same. Find equivalent ratios where the lemon is the same in both jugs.

  • Ana: \(2 : 3 = 6 : 9\) (times 3).
  • Ben: \(3 : 5 = 6 : 10\) (times 2).

Now both jugs have 6 cups of lemon, but Ben's has 10 cups of water against Ana's 9. Ben's lemon is spread through more water, so Ana's is stronger.

Way 2: compare the fractions of the whole.

  • Ana: lemon is \(\tfrac{2}{2 + 3} = \tfrac25\) of the drink.
  • Ben: lemon is \(\tfrac{3}{3 + 5} = \tfrac38\) of the drink.

Using 40 as a common bottom: \(\tfrac25 = \tfrac{16}{40}\) and \(\tfrac38 = \tfrac{15}{40}\). Ana's is a little stronger. Both ways agree, as they must.

Two mixes taste the same exactly when their ratios are equivalent, that is, when they simplify to the same simplest form.

Worked examples

1. Simplify \(24 : 36\).

  • Both are even: \(12 : 18\). Still both even: \(6 : 9\). Both divide by 3: \(2 : 3\).
  • Or in one step: the greatest common factor of 24 and 36 is 12, and \(24 \div 12 = 2\), \(36 \div 12 = 3\).
  • Answer: \(2 : 3\).

2. Are \(15 : 25\) and \(9 : 15\) the same mix?

  • \(15 : 25\): divide by 5 to get \(3 : 5\).
  • \(9 : 15\): divide by 3 to get \(3 : 5\).
  • Same simplest form, so yes, they are equivalent.

3. Mixing paint. A green paint is made with blue : yellow \(= 2 : 5\). How much yellow goes with 12 litres of blue?

  • To get from 2 to 12 we multiply by 6.
  • Multiply the other part by 6 too: \(5 \times 6 = 30\) litres of yellow.
  • Check: \(12 : 30\) divides by 6 to give \(2 : 5\). ✓

4. Filling a ratio table. Flour : sugar \(= 3 : 2\). Fill in the gaps.

Flour (cups) 3 6 ? 15 4.5
Sugar (cups) 2 ? 8 ? ?
  • \(6 = 3 \times 2\), so sugar is \(2 \times 2 = 4\).
  • \(8 = 2 \times 4\), so flour is \(3 \times 4 = 12\).
  • \(15 = 3 \times 5\), so sugar is \(2 \times 5 = 10\).
  • \(4.5 = 3 \times 1.5\), so sugar is \(2 \times 1.5 = 3\).

5. Which class has more girls for its size? Class A has girls : boys \(= 4 : 5\). Class B has \(5 : 6\).

  • Make the girls the same: \(4 : 5 = 20 : 25\) and \(5 : 6 = 20 : 24\).
  • For 20 girls, class A has 25 boys and class B has only 24. So class B has a slightly larger share of girls.
  • Check with fractions: A is \(\tfrac49\), B is \(\tfrac{5}{11}\). As decimals, \(\tfrac49 \approx 0.444\) and \(\tfrac{5}{11} \approx 0.455\). B is bigger. ✓

Try it

  1. Simplify \(20 : 35\).
  2. Simplify \(45 : 60\).
  3. Write \(2.5 : 4\) in simplest form with whole numbers.
  4. Are \(6 : 8\) and \(9 : 12\) equivalent? Show how you know.
  5. A cordial is mixed cordial : water \(= 1 : 6\). How much water goes with 250 mL of cordial?
  6. Simplify \(10 : 15 : 25\).
  7. Write \(\tfrac12 : \tfrac34\) in simplest form.
  8. Orange paint A is red : yellow \(= 3 : 4\). Orange paint B is red : yellow \(= 5 : 7\). Which is redder?
Answers
  1. Both divide by 5: \(4 : 7\).
  2. Both divide by 15: \(3 : 4\). (In steps: by 5 gives \(9 : 12\), then by 3 gives \(3 : 4\).)
  3. Multiply both by 2: \(5 : 8\). No number bigger than 1 divides both 5 and 8, so that is the simplest form.
  4. \(6 : 8 = 3 : 4\) (divide by 2) and \(9 : 12 = 3 : 4\) (divide by 3). Same simplest form, so yes.
  5. 250 is \(1 \times 250\), so water is \(6 \times 250 = 1500\) mL (1.5 litres).
  6. All three divide by 5: \(2 : 3 : 5\).
  7. Multiply both by 4: \(\tfrac12 \times 4 = 2\) and \(\tfrac34 \times 4 = 3\), so \(2 : 3\).
  8. Make the red the same: \(3 : 4 = 15 : 20\) and \(5 : 7 = 15 : 21\). For 15 red, A has less yellow, so A is redder. Check with fractions: A is \(\tfrac37 = \tfrac{36}{84}\) red and B is \(\tfrac{5}{12} = \tfrac{35}{84}\) red, so A is redder. ✓

Common mistake

Adding instead of multiplying. "The recipe is 2 eggs to 3 cups of flour; I have 5 cups of flour, so I add 2 to both and use 4 eggs." No: \(4 : 5\) is not the same mix as \(2 : 3\). From 3 cups to 5 cups is multiplying by \(\tfrac53\), so the eggs should be \(2 \times \tfrac53 = \tfrac{10}{3}\), about 3 and a third eggs.

Changing only one part. To simplify \(12 : 18\), dividing just the 12 by 6 gives \(2 : 18\), a completely different mix. Whatever you do to one part, do to the other.

Where it leads