Number · Ratios and Proportion · Lesson 2

Unit Rates and Rate Language

"GH₵9 a litre", "60 kilometres per hour", "GH₵20 per kilo": rates are everywhere in Ghanaian life, from the fuel station to the trotro to the meat seller at Salaga Market. In this lesson you will turn a ratio into a unit rate (the amount for ONE unit) and use it to work out costs, distances and best buys quickly and correctly.

NaCCA B7.1.4.1.2 ⏱ about 45 min

By the end you will be able to

✔Explain the difference between a ratio and a rate.
✔Write a ratio a : b as a unit rate $\frac{a}{b}$ per one unit of the second quantity.
✔Use unit rates (the unitary method) to find costs, distances and times.
✔Complete tables of values using a unit rate and compare rates to find the better buy.

Key words

Rate: A comparison of two quantities measured in different units, e.g. GH₵18 for 2 litres or 120 km in 2 hours.
Unit rate: A rate for ONE unit of the second quantity, e.g. GH₵9 per litre or 60 km per hour.
Per: "For each" or "for every one". km/h means kilometres per hour.
Unitary method: Find the value of one unit first, then multiply to find the value of any number of units.
Unit price: The cost of one item or one unit of measure, used to compare prices.

Curriculum reference

Content standard: B7.1.4.1 Demonstrate an understanding of the concept of ratios and its relationship to fractions and use it to solve problems that involve rates, ratios, and proportional reasoning.

Indicator: Use the concept of a unit rate associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio relationship.

Starter

Which shop is cheaper?

Selorm's mother sends him to Kejetia to buy rice. Shop A sells a 2 kg bag for GH₵44. Shop B sells a 5 kg bag for GH₵105. Shop B's price looks bigger, but its bag is bigger too! How can Selorm decide which shop gives better value? The secret is to find the price of just ONE kilogram in each shop. That single number is called a unit rate, and by the end of this lesson you will be able to find it in seconds.

1. From ratio to rate

In the last lesson, a ratio compared quantities of the same kind, like boys to girls. A rate compares quantities measured in different units: cedis and litres, kilometres and hours, square yards and minutes. Because the units are different, we always write them in the answer.

Ratio (same kind)	Rate (different units)
12 boys : 18 girls	GH₵18 for 2 litres of drink
1 headpan cement : 3 headpans sand	120 km in 2 hours
no units in the answer	units in the answer: GH₵ per litre, km per hour

A unit rate tells you how much of the first quantity there is for one unit of the second. For a ratio a : b (with b not zero), the unit rate is $\frac{a}{b}$ of the first quantity per one of the second. For example, a recipe with 3 cups of flour to 4 cups of sugar has $\frac{3}{4}$ cup of flour for each cup of sugar. Aisha polishes 8 square yards of floor tiles every 7 minutes, so she polishes $\frac{8}{7}$ square yards per minute (a little more than 1 square yard each minute).

2. Finding a unit rate

1Decide what you want "per one" of: per litre, per hour, per kilogram, per minute.
2Divide the other quantity by the number of those units.
3Write the answer with both units, using the word "per" (or a slash, like km/h).

Information	Division	Unit rate
A trotro travels 120 km in 2 hours	120 ÷ 2	60 km per hour
2 litres of Coca-Cola cost GH₵18	18 ÷ 2	GH₵9 per litre
3 kg of meat cost GH₵60	60 ÷ 3	GH₵20 per kg
Kofi earns GH₵240 for 8 hours	240 ÷ 8	GH₵30 per hour
A tap fills 40 litres in 5 minutes	40 ÷ 5	8 litres per minute

3. Using a unit rate to solve problems

Once you know the value of one unit, you can find the value of any number of units by multiplying. This is the unitary method. If 2 litres of Coca-Cola cost GH₵18, one litre costs GH₵9, so 3 litres cost 3 × 9 = GH₵27, 7 litres cost 7 × 9 = GH₵63, and even 1.5 litres cost 1.5 × 9 = GH₵13.50.

▶ Each jump of 1 litre adds the same GH₵9. That constant jump is the unit rate.

Tables are a neat way to organise rates. At Salaga Market, 3 kg of meat costs GH₵60. The unit rate is GH₵20 per kg, so every cost in the table is 20 times the mass. Step through the animation to fill the table.

▶ Find the cost of 1 kg first; then every cost in the table is 20 × the mass.

4. Comparing rates: the better buy

Back to Selorm at Kejetia. To compare two prices fairly, find the unit price (cost per kg) in each shop. The smaller unit price is the better buy.

Shop	Offer	Unit price
A	2 kg for GH₵44	44 ÷ 2 = GH₵22 per kg
B	5 kg for GH₵105	105 ÷ 5 = GH₵21 per kg

Shop B is cheaper by GH₵1 per kilogram, even though its total price is bigger. Rate language helps you explain your answer clearly: "Shop B charges GH₵21 per kilogram, which is less than Shop A's GH₵22 per kilogram."

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Writing a ratio as a unit rate

A recipe has a ratio of 3 cups of flour to 4 cups of sugar. How much flour is there for each cup of sugar?

Show the solution

Step 1We want flour per ONE cup of sugar, so divide the flour by the sugar.
Step 2Unit rate = $\frac{3}{4}$ cup of flour per cup of sugar.
Step 3Check: 4 cups of sugar × $\frac{3}{4}$ = 3 cups of flour ✓

Answer: $\frac{3}{4}$ cup of flour for each cup of sugar

Example 2

Speed of a trotro

A trotro travels 150 km from Accra towards Kumasi in 3 hours. Find its average speed, and how far it would travel in 5 hours at the same speed.

Show the solution

Step 1Speed = 150 ÷ 3 = 50 km per hour.
Step 2Distance in 5 hours = 5 × 50 = 250 km.

Answer: 50 km/h; 250 km

Example 3

Costs of Coca-Cola

If 2 litres of Coca-Cola cost GH₵18, find the cost of (a) 1.5 litres (b) 3 litres (c) 7 litres.

Show the solution

Step 1Unit rate: 18 ÷ 2 = GH₵9 per litre.
Step 2(a) 1.5 × 9 = GH₵13.50
Step 3(b) 3 × 9 = GH₵27
Step 4(c) 7 × 9 = GH₵63

Answer: (a) GH₵13.50 (b) GH₵27 (c) GH₵63

Example 4

Completing a table at Salaga Market

3 kg of meat costs GH₵60. Complete the costs for 2 kg, 5 kg and 12 kg.

Show the solution

Step 1Unit rate: 60 ÷ 3 = GH₵20 per kg.
Step 22 kg: 2 × 20 = GH₵40.
Step 35 kg: 5 × 20 = GH₵100.
Step 412 kg: 12 × 20 = GH₵240.

Answer: GH₵40, GH₵100, GH₵240

Example 5

Finding the better buy

Shop A sells 2 kg of rice for GH₵44 and Shop B sells 5 kg for GH₵105. Which is the better buy?

Show the solution

Step 1Shop A: 44 ÷ 2 = GH₵22 per kg.
Step 2Shop B: 105 ÷ 5 = GH₵21 per kg.
Step 3GH₵21 < GH₵22, so Shop B is cheaper per kilogram.

Answer: Shop B (GH₵21 per kg)

Summary

A rate compares quantities in different units; always write the units, e.g. km per hour.
Unit rate = first quantity ÷ second quantity: the amount for ONE unit (a : b gives $\frac{a}{b}$ per one).
Unitary method: find the value of one unit, then multiply for any number of units.
Divide by the unit rate to go backwards (how many units for a given amount).
Compare unit prices to find the better buy: the smaller unit price wins.

Which shop is cheaper?

1. From ratio to rate

2. Finding a unit rate

3. Using a unit rate to solve problems

4. Comparing rates: the better buy

Worked examples

Writing a ratio as a unit rate

Speed of a trotro

Costs of Coca-Cola

Completing a table at Salaga Market

Finding the better buy

Summary

Practice: 15 questions