For every a, there are b

The idea

A ratio a : b says that for every a units of the first quantity there are b units of the second quantity; a and b are called the terms of the ratio.

Manjunath's filter coffee

Manjunath mixes 15 mL of coffee decoction with 35 mL of milk to make one cup of filter coffee. We say the ratio of decoction to milk is 15 : 35.

For two customers he pours the recipe two times into a jug. What does the jug hold?

• 30 mL decoction and 70 mL milk
• 30 mL decoction and 35 mL milk
• 15 mL decoction and 70 mL milk

Pour the sets

One set is the recipe: decoction and milk. Pour sets into the jug. Look at how the dark and cream parts repeat. Try all three coffees.

For every a, there are b

A ratio a : b says that for every a units of the first quantity there are b units of the second quantity; a and b are called the terms of the ratio.

In 15 : 35, the first term is 15 (decoction) and the second term is 35 (milk). Order matters: the ratio of milk to decoction is 35 : 15.

Notes

A ratio a : b says that for every a units of the first quantity there are b units of the second quantity; a and b are called the terms of the ratio.

Check yourself

Stronger coffee has the ratio 20 : 30 of decoction to milk. Which sentence is right?

Lighter coffee is 10 : 40 (decoction to milk). A jug is made with 3 sets. How many mL of milk does it hold?

Answer: 120 mL

One set has 40 mL of milk, so 3 sets have 40 × 3 = 120 mL.

In 20 : 30 (decoction to milk), what is the first term and what does it measure?

A jug holds 30 mL of decoction and 70 mL of milk. What is the ratio of milk to decoction?

• For every 20 mL of milk there are 30 mL of decoction. The order is the other way round. The first term, 20, is decoction.
• For every 20 mL of decoction there are 30 mL of milk — correct. Yes! The first term goes with the first quantity, decoction, and the second term with milk.
• There are 50 mL of decoction for every 30 mL of milk. That adds the terms. The ratio says 20 mL of decoction goes with 30 mL of milk.
• 30, the mL of decoction. The first term is 20. And 30 measures the milk, the second quantity.
• 20, the mL of milk. 20 is the first term, but it goes with the first quantity, which is decoction.
• 20, the mL of decoction — correct. Yes! The first term, 20, goes with the first quantity, decoction.
• 50, the mL in the whole set. 50 is the total of both terms. A term is one of the two numbers of the ratio.
• 30 : 70. That is decoction to milk. We are asked for milk to decoction, so milk comes first.
• 70 : 30 — correct. Yes! Milk is the first quantity, so 70 is the first term.
• 100 : 30. 100 is the total of the jug. A term is the amount of one quantity.
• 70 : 100. The second quantity is decoction, 30 mL, not the total.