Math Detours · Grade 6 · Sharpen · 2

Ratios, rates and recipes

A ratio is a recipe. Once you can scale a recipe up and down without thinking, speed, prices, maps and mixing problems are all the same problem.

Name ____________________Date ____________

Tonight's idea

A ratio compares amounts. Pancakes might need flour and milk in the ratio 3 : 2, meaning 3 cups of flour for every 2 of milk. The ratio stays the same whether you cook for two people or two hundred. 3 : 2, 6 : 4, 300 : 200 are all the same recipe.

A rate is a ratio between two different kinds of thing: kilometres and hours, dollars and pens. A unit rate is the rate per one: 72 km per hour, 65 cents per pen. Unit rates let you compare anything. To compare two deals or two speeds, bring both to "per one".

The one tool you need is the scale factor. If 4 people need 300 g of flour, then 10 people need \(300 \times \tfrac{10}{4} = 750\) g. The scale factor \(\tfrac{10}{4} = 2.5\) multiplies every ingredient. Do not add; multiply.

Problems about taps filling baths, trains meeting, and painters painting houses together all come down to adding rates, never adding times. A tap that fills a bath in 12 minutes fills \(\tfrac1{12}\) of the bath per minute; a tap that takes 4 minutes fills \(\tfrac14\). Together: \(\tfrac1{12} + \tfrac14 = \tfrac13\) per minute, so 3 minutes. Rates add; times do not.

Watch one

A train covers 180 km in 2.5 hours. How far does it go in 4 hours 15 minutes at the same speed?

Unit rate first: \(180 \div 2.5 = 72\) km per hour. Then \(4\) h \(15\) min is \(4.25\) h, so distance \(= 72 \times 4.25 = 288 + 18 = 306\) km.

Alternatively, a ratio table: 2.5 h → 180 km; 0.5 h → 36 km; 4 h → 288 km; 0.25 h → 18 km; total 306 km.

Your turn

  1. Pens: 6 for $4.50 or 10 for $7.00. Which is the better deal, and by how much per pen?
  2. Paint is mixed red : blue = 3 : 5. (a) How much of each in 24 litres? (b) If you have 12 litres of red, how much blue do you need?
  3. A map has scale 1 : 50 000. Two towns are 7 cm apart on the map. How far apart are they really, in km? (Hint: 1 km = 100 000 cm.)
  4. Three friends split $84 in the ratio 2 : 3 : 7. How much does each get?
  5. A pancake recipe for 4 people uses 300 g flour, 2 eggs and 350 ml milk. Rewrite it for 10 people, then for 3 people.
  6. One tap fills a bath in 12 minutes, another in 6 minutes. With both running, how long does it take? (Think about what fraction of the bath each fills per minute. The answer is not 9 minutes.)

Stretch: mixing juices

Juice A is 20% real fruit. Juice B is 50% real fruit. (a) Mix 2 litres of A with 3 litres of B. What percentage of the mixture is real fruit? (b) Now the harder one: how many litres of B must you add to 2 litres of A to get a mixture that is exactly 40% fruit?

Keep track of the litres of real fruit, not the percentages. Percentages do not add; litres do.

Answers

  1. $4.50 ÷ 6 = 75 cents per pen; $7.00 ÷ 10 = 70 cents per pen. The 10-pack is better by 5 cents a pen.
  2. (a) 8 parts in total, so each part is 3 L: 9 L red, 15 L blue. (b) 12 L of red is 4 L per part, so 20 L of blue.
  3. 7 cm × 50 000 = 350 000 cm = 3.5 km.
  4. 12 parts, each $7: $14, $21, $49.
  5. Scale factor 2.5: 750 g flour, 5 eggs, 875 ml milk. Scale factor 0.75: 225 g flour, 1.5 eggs (crack two, use one and a half, or accept a slightly eggier pancake), 262.5 ml milk.
  6. Per minute: \(\tfrac1{12} + \tfrac16 = \tfrac1{12} + \tfrac2{12} = \tfrac3{12} = \tfrac14\) of the bath. So 4 minutes. Together they are faster than the faster tap alone, which 9 minutes could never be.

Stretch. (a) Fruit: \(0.2 \times 2 + 0.5 \times 3 = 0.4 + 1.5 = 1.9\) L out of 5 L, so 38%. (b) Adding \(x\) litres of B: fruit \(= 0.4 + 0.5x\), total \(= 2 + x\). We need \(0.4 + 0.5x = 0.4(2 + x) = 0.8 + 0.4x\), so \(0.1x = 0.4\) and \(x = 4\) L. Check: 2.4 L fruit in 6 L is 40%. A quicker way: 40% is twice as far from 20% as from 50% (20 away versus 10 away), so you need twice as much B as A.

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