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B2.8 · Compare proportional situations and find unknown values

Learn to compare proportional situations and find unknown values through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Number

Keep matching amounts together and find unknown values

A ratio compares two amounts in a chosen order. For example, a hypothetical recipe uses 2 cups of oats for 6 snacks. Those amounts make a pair because they belong together. If the recipe stays proportional, changing the number of snacks changes the oats by the same factor. In this lesson, you will compare proportional situations and find unknown values. All examples use hypothetical information for practice.

What you will learn

Recognize when paired amounts keep the same ratio.
Compare proportional situations using matching units.
Find an unknown amount by scaling a pair or using a unit rate.

1. Understand proportional pairs

A pair is two amounts that belong together. In the recipe, 2 cups of oats and 6 snacks make one pair. If the number of snacks doubles, the oats must double too: 4 cups for 12 snacks.

A ratio compares two amounts in a stated order. For example, you can compare cups of oats with snacks. Keep that order the same when you compare pairs. The units help you remember what each number means.

Paired amounts are proportional when they keep the same ratio. One way to check is to see whether both amounts are multiplied or divided by the same number. That number is called the scale factor. A scale factor describes how one pair changes into another pair.

A table keeps matching amounts clear. Read across a row to see which values belong together. In the recipe table, each increase in snacks is matched by the same change in oats per snack. The amounts in the table are hypothetical.

The ratio in the first pair matches the ratio in the second pair. Written with numbers, the comparison is

\frac{2}{6}=\frac{4}{12}

. Both fractions describe cups of oats per snack. The equal fractions show that the pairs are proportional.

A ratio compares amounts in a stated order.
Proportional pairs keep the same ratio.
A scale factor multiplies or divides both amounts in a pair.

2. Compare situations fairly

Before comparing situations, name the quantities and their units. Compare the same kinds of quantities. For example, kilometres per hour can be compared with kilometres per hour. Comparing just the distances would not be fair if the travel times differ.

A unit rate tells how much of one quantity goes with 1 unit of another. To find a unit rate, divide the amount by the number of units. If a hypothetical route covers 12 kilometres in 3 hours, divide 12 by 3. The route covers 4 kilometres in 1 hour.

Suppose another hypothetical route covers 15 kilometres in 5 hours. It covers 3 kilometres in 1 hour. Since both rates use kilometres per hour, you can compare them directly. The first route covers more kilometres in one hour.

You can also compare situations at the same number of units. Finding each route's distance in 1 hour is one way to do this. A table can help keep the amounts and units matched. For a fair comparison, calculate the same kind of rate for each situation.

Order matters when finding a rate. For kilometres per hour, divide kilometres by hours. Reversing the order gives hours per kilometre, which answers a different question. Always check that your result has the units you need.

A unit rate gives an amount for 1 unit of another quantity.
Use matching units to compare fairly.
Keep the quantity order consistent.

3. Find an unknown value

First identify what each number counts. Keep the units and order clear. Then decide how the known amounts change to reach the amounts you need. If you multiply one amount in a proportional pair, multiply the other amount by the same factor.

For example, if 3 metres of ribbon make 8 decorations, then 16 decorations is twice as many as 8. The ribbon amount must also double. This gives 6 metres.

If the numbers do not change by an easy whole-number factor, find the amount for one unit. Divide the known amount by the number of units. Then multiply that amount by the number of units needed. This works because the amount for each one unit stays the same in a proportional situation.

Check your result by comparing the original pair with the completed pair. Both quantities should have changed by the same factor. Ask whether the answer makes sense for the situation. Include the correct unit in your answer.

A table can show an unknown clearly. Put the known pair in one row and the new situation in another. Write the known amounts in the right columns, then use scaling or a unit rate to fill the empty place. Do not compare numbers that count different things.

Scale both quantities by the same factor.
A unit rate can help when direct scaling is not easy.
Check the units and the meaning of the answer.

4. Use a clear solving routine

Name the two quantities and their units. Put matching amounts together, using a table if it helps. Read each pair in the same order.

For a comparison, find unit rates or compare both situations at the same number of units. For an unknown value, scale a known pair or find the amount for one unit and scale from there.

Try this practice question: A hypothetical design uses 4 metres of cord for 6 items. How much cord is needed for 18 items if every item uses the same amount? The item count triples, so the cord amount must also triple. The answer is 12 metres.

Before you finish, ask yourself: Did I compare the same quantities? Did I use matching units? Did both amounts change by the same factor? Does my answer fit the situation? These checks can catch a reversed ratio or a calculation based on mismatched quantities.

4\times3=12

Organize paired values before calculating.
Use matching units for a fair comparison.
Check the ratio and the answer in context.

Paired amounts in the hypothetical recipe

Oats (cups)	Snacks
2	6
4	12
6	18

Worked example

Scale a recipe

A hypothetical recipe uses 3 cups of oats for 12 snacks. How many cups are needed for 20 snacks if the recipe stays proportional?

Oats for one snack

One snack uses one quarter cup of oats.

Find oats for one snack
Divide the cups of oats by the number of snacks. This gives the amount of oats for 1 snack.
$3\div12=0.25$
Find oats for 20 snacks
Multiply the amount for one snack by 20. The result is in cups because the rate is cups per snack.
$0.25\times20=5$

Answer: The recipe needs 5 cups of oats for 20 snacks.

Check: The original pair gives 0.25 cup per snack. The completed pair also gives 5 divided by 20, or 0.25 cup per snack.

Worked example

Compare two routes

Two hypothetical routes are described. Route A covers 18 kilometres in 3 hours. Route B covers 20 kilometres in 4 hours. Which route covers more kilometres per hour?

Distance covered in one hour

Hypothetical route rates, measured in kilometres per hour.

Find Route A's unit rate
Divide the distance in kilometres by the time in hours. This gives the distance covered in 1 hour.
$18\div3=6$
Find Route B's unit rate
Use the same order and units for Route B so the rates can be compared fairly.
$20\div4=5$
Compare the unit rates
Both rates are in kilometres per hour. Since 6 is greater than 5, Route A covers more distance in one hour.
$6>5$

Answer: Route A covers more: 6 kilometres per hour compared with 5 kilometres per hour for Route B.

Check: Both rates describe kilometres in 1 hour, so the comparison uses matching units.

Worked example

Complete a proportional pair

A hypothetical craft project uses 5 metres of cord for 8 items. How many metres are needed for 24 items if each item uses the same amount?

Cord and item pairs

Hypothetical proportional pairs: 8 items use 5 metres, and 24 items use 15 metres.

Find the scale factor
The number of items changes from 8 to 24. Divide the new number by the original number to find the factor.
$24\div8=3$
Scale the cord amount
The cord amount must be multiplied by the same factor because each item uses the same amount of cord.
$5\times3=15$
Check the pair
Both amounts were multiplied by 3, so the ratio of metres of cord to items stays the same.
$\frac{5}{8}=\frac{15}{24}$

Answer: The project needs 15 metres of cord for 24 items.

Check: There are three times as many items, and three times 5 metres is 15 metres.

Common mistakes and how to avoid them

Comparing 18 kilometres with 20 kilometres without considering the different times.

Correction: Find the kilometres per hour for each route, then compare those matching rates.

Multiplying only one amount when scaling a proportional pair.

Correction: Multiply or divide both amounts by the same factor.

Switching the order of quantities in a ratio.

Correction: Keep the same order in each pair, such as cups of oats compared with snacks.

Lesson summary

Proportional pairs keep the same ratio.
Scale both amounts by the same factor to find a matching value.
A unit rate gives an amount for 1 unit and helps compare situations fairly.
Keep units and quantity order consistent, then check that the answer fits.

Check your understanding

Question 1

A hypothetical machine fills 14 bottles in 2 minutes at a steady rate. How many bottles does it fill in 1 minute?

6 bottles
7 bottles
12 bottles
16 bottles

Show answer and explanation

7 bottles

Divide 14 bottles by 2 minutes. The machine fills 7 bottles per minute.

Question 2

Situation A uses 4 metres of fabric for 10 bags. Situation B uses 8 metres for 20 bags. Are the situations proportional?

Yes, because both amounts in Situation B are twice the amounts in Situation A.
No, because Situation B uses more fabric.
No, because the numbers of bags are different.
Yes, because the fabric amounts differ by 4 metres.

Show answer and explanation

Yes, because both amounts in Situation B are twice the amounts in Situation A.

Both the fabric amount and the number of bags are multiplied by 2. The pairs keep the same ratio.

Question 3

A hypothetical mixture uses 6 cups of water for 9 cups of another ingredient. How much water is needed for 3 cups of the other ingredient?

1 cup
2 cups
3 cups
6 cups

Show answer and explanation

2 cups

The other ingredient is divided by 3, from 9 cups to 3 cups. Divide the water amount by 3 as well: 6 divided by 3 is 2 cups.

Key terms

Ratio: A comparison of two amounts in a stated order.
Proportional: Describes paired amounts that keep the same ratio.
Scale factor: A number used to multiply or divide amounts when scaling a pair.
Unit rate: An amount of one quantity for 1 unit of another quantity.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.8. It is a study resource, not an official curriculum publication.

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