🎬 Keep equivalent ratios balanced
Multiply or divide every part of a ratio by the same number.
🎯 By the end of this lesson...
You should be able to:
- Explain what a ratio represents.
- Write ratios in the correct order.
- Simplify ratios fully.
- Find equivalent ratios.
- Convert quantities to the same units before comparing them.
- Share a quantity in a given ratio.
- Find a missing quantity using a ratio.
- Explain what a rate is.
- Calculate unit rates.
- Solve direct-proportion problems.
- Solve inverse-proportion problems.
- Apply ratio, rate and proportion in practical situations.
🤔 Think About This...
Mixing fruit juice
Tariro makes fruit juice using:
- 1 cup of concentrate;
- 4 cups of water.
If she uses 3 cups of concentrate, how much water should she use to keep the same taste?
1 : 4 becomes 3 : 12.
1️⃣ What Is a Ratio?
A ratio compares the amount of one quantity with the amount of another quantity.
This is read as:
It means that for every 3 parts of the first quantity, there are 5 parts of the second quantity.
2️⃣ The Order of a Ratio Matters
The quantities in the ratio must be written in the same order in which they are mentioned.
Example 1
A class contains 12 boys and 18 girls.
Boys : Girls
Girls : Boys
💡 Did You Know?
Ratios appear in many everyday activities.
3️⃣ Writing Ratios
A comparison may be written in several forms.
Using a colon
Using words
As a fraction
4️⃣ Simplifying Ratios
Ratios are simplified by dividing every term by the same common factor.
Example 2: Simplify 12 : 18
Find the HCF of 12 and 18.
🧠 StudyNest Method: Simplifying a Ratio
Step 2: Find the HCF of all the terms.
Step 3: Divide every term by the HCF.
Step 4: Check that the terms no longer have a common factor greater than 1.
Step 5: Keep the quantities in their original order.
5️⃣ Ratios with Different Units
Quantities must use the same units before they can be compared in a ratio.
Example 3: Simplify 2 m : 50 cm
Convert 2 metres to centimetres.
Divide both terms by 50.
6️⃣ Ratios Containing Decimals
Example 4: Simplify 1.5 : 2.5
Multiply both terms by 10 to remove the decimals.
Divide both terms by 5.
7️⃣ Ratios Containing Fractions
Example 5: Simplify 1/2 : 3/4
Multiply both terms by the LCM of the denominators.
8️⃣ Equivalent Ratios
Equivalent ratios describe the same comparison using different numbers.
9️⃣ Finding a Missing Term in a Ratio
Example 6: Find x
Ask how 3 became 12.
Multiply the second term by the same number.
🔟 Sharing a Quantity in a Given Ratio
Example 7: Divide US$120 in the ratio 2 : 3
First find the total number of ratio parts.
Find the value of one part.
Calculate each share.
Check
🧠 StudyNest Method: Sharing in a Ratio
Step 2: Divide the total quantity by the total number of parts.
Step 3: Multiply the value of one part by each term in the ratio.
Step 4: Include the correct units.
Step 5: Add the shares to check that they equal the original total.
🌍 Real-Life Example: Sharing Business Profit
Example 8
Chipo and Nyasha invest US$300 and US$500 respectively in a small business.
They share a profit of US$240 according to their investments.
Their investment ratio is:
Total ratio parts:
Value of one part:
1️⃣1️⃣ What Is a Rate?
A rate compares two quantities measured in different units.
Speed
Wages
Fuel consumption
Unit price
1️⃣2️⃣ Finding a Unit Rate
A unit rate tells us the amount for one unit.
Example 9: Find the cost per book
Five exercise books cost US$15.
Example 10: Calculate the average speed
A car travels 180 km in 3 hours.
1️⃣3️⃣ Comparing Rates
Example 11: Which packet offers better value?
Packet A
4 kg for US$10
Packet B
6 kg for US$13.20
1️⃣4️⃣ What Is Proportion?
A proportion states that two ratios are equal.
Both ratios simplify to 2 : 5.
1️⃣5️⃣ Direct Proportion
Two quantities are in direct proportion when they increase or decrease in the same ratio.
🧠 StudyNest Method: Direct Proportion
Step 2: Find the value of one unit.
Step 3: Multiply the unit value by the required number of units.
Step 4: Include the correct units.
Step 5: Check that a larger quantity produces a proportionally larger result.
1️⃣6️⃣ Solving Direct-Proportion Problems
Example 12
Three shirts cost US$45.
Find the cost of eight shirts at the same price.
First find the cost of one shirt.
Then find the cost of eight shirts.
1️⃣7️⃣ Direct Proportion Using an Equation
If y is directly proportional to x, we write:
This becomes:
where k is the constant of proportionality.
Example 13
y is directly proportional to x.
When x = 4, y = 20.
Find y when x = 7.
Therefore:
1️⃣8️⃣ Inverse Proportion
Two quantities are in inverse proportion when one increases while the other decreases in such a way that their product remains constant.
More time needed
Less time needed
In inverse proportion, they move in opposite directions.
🧠 StudyNest Method: Inverse Proportion
Step 2: Multiply the first pair of corresponding values.
Step 3: Keep that product constant.
Step 4: Divide the constant by the new known value.
Step 5: Check that the result moves in the expected opposite direction.
1️⃣9️⃣ Solving an Inverse-Proportion Problem
Example 14
Six workers complete a job in 10 days.
Assuming they work at the same rate, how many days would 12 workers take?
Double the workers should halve the time.
2️⃣0️⃣ Inverse Proportion Using an Equation
If y is inversely proportional to x, we write:
This becomes:
Example 15
y is inversely proportional to x.
When x = 3, y = 8.
Find y when x = 6.
Therefore:
⚖️ Direct and Inverse Proportion
Direct Proportion
Both quantities move in the same direction.
If x doubles, y doubles.
Inverse Proportion
The quantities move in opposite directions.
If x doubles, y halves.
🌍 Real-Life Example: Recipe Proportion
Example 16
A recipe for four people requires 300 g of flour.
How much flour is required for ten people?
Flour for one person:
Flour for ten people:
🌍 Real-Life Example: Fuel Rate
Example 17
A vehicle travels 420 km using 35 litres of fuel.
Find the distance travelled per litre.
🧠 The StudyNest Ratio and Proportion Strategy
Step 2: Check the order and units.
Step 3: Decide whether the question involves:
simplifying, sharing, a rate, direct proportion or inverse proportion.
Step 4: Select the correct method.
Step 5: Calculate one clear stage at a time.
Step 6: Include the correct units.
Step 7: Check whether the answer behaves as expected.
🎯 Exam Tips
⚠ Common Mistakes
Reversing the order of the ratio.
Comparing quantities that use different units.
Dividing only one term when simplifying a ratio.
Sharing the total using the individual ratio numbers without first finding the total number of parts.
Forgetting to include units in a rate.
Treating inverse proportion as direct proportion.
Assuming that any two quantities increasing together must be exactly proportional.
🎮 Your Turn!
🟢 Foundation Questions
-
Simplify:
8 : 12 -
Simplify:
15 : 25 -
Simplify:
18 : 24 : 30 -
Simplify:
3 m : 75 cm -
Simplify:
1.2 : 2 -
Find x:
2 : 7 = 8 : x - Write an equivalent ratio to 3 : 5 by multiplying both terms by 4.
🔵 Developing Questions
- Share US$150 in the ratio 2 : 3.
- Divide 84 kg in the ratio 3 : 4.
- Divide 360 sweets in the ratio 2 : 3 : 4.
- Four books cost US$18. Find the cost per book.
- A vehicle travels 240 km in 4 hours. Find its average speed.
- Five kilograms of rice cost US$12.50. Find the cost per kilogram.
- Three notebooks cost US$6. Find the cost of eight notebooks at the same rate.
- A recipe for six people requires 450 g of flour. Find the amount required for ten people.
🔴 Challenge Questions
- Two partners invest US$400 and US$600 in a business. Share a profit of US$350 according to their investments.
- y is directly proportional to x. When x = 6, y = 24. Find y when x = 15.
- p is directly proportional to q. When p = 35 and q = 5, find the equation connecting p and q.
- Eight workers complete a job in 15 days. How long would 12 workers take at the same rate?
- y is inversely proportional to x. When x = 4, y = 12. Find y when x = 8.
- Six taps fill a tank in 20 minutes. How long would eight identical taps take?
- A map uses a scale of 1 : 50 000. What actual distance is represented by 6 cm on the map?
- The ratio of boys to girls in a school is 7 : 8. If there are 360 learners altogether, how many are girls?
✅ Show Answers
- 8 : 12 = 2 : 3
- 15 : 25 = 3 : 5
- 18 : 24 : 30 = 3 : 4 : 5
-
3 m = 300 cm
300 : 75 = 4 : 1 -
1.2 : 2
= 12 : 20
= 3 : 5 -
2 × 4 = 8
7 × 4 = 28
x = 28 -
3 × 4 : 5 × 4
= 12 : 20 -
Total parts = 5
One part = 150 ÷ 5 = 30
Shares = US$60 and US$90 -
Total parts = 7
One part = 84 ÷ 7 = 12
Shares = 36 kg and 48 kg -
Total parts = 9
One part = 360 ÷ 9 = 40
Shares = 80, 120 and 160 sweets - 18 ÷ 4 = US$4.50 per book
- 240 ÷ 4 = 60 km/h
- 12.50 ÷ 5 = US$2.50 per kg
-
Cost per notebook = 6 ÷ 3 = 2
Cost of eight = 8 × 2
= US$16 -
Flour per person = 450 ÷ 6 = 75 g
For ten people = 75 × 10
= 750 g -
Investment ratio:
400 : 600 = 2 : 3
Total parts = 5
One part = 350 ÷ 5 = 70
Shares = US$140 and US$210 -
y = kx
24 = 6k
k = 4
y = 4 × 15
= 60 -
p = kq
35 = 5k
k = 7
Equation: p = 7q -
Workers × days = constant
8 × 15 = 120
Days = 120 ÷ 12
= 10 days -
xy = constant
4 × 12 = 48
y = 48 ÷ 8
= 6 -
Taps × time = constant
6 × 20 = 120
Time = 120 ÷ 8
= 15 minutes -
Actual distance:
6 × 50 000 = 300 000 cm
300 000 cm = 3 km -
Total parts = 7 + 8 = 15
One part = 360 ÷ 15 = 24
Girls = 8 × 24
= 192 girls
🧠 Remember This
The order of a ratio matters.
Convert quantities to the same units before comparing them.
Simplify a ratio by dividing every term by the HCF.
When sharing, add the ratio parts first.
A rate compares quantities measured in different units.
Direct proportion means both quantities change in the same direction.
Inverse proportion means one quantity increases while the other decreases.
🎯 Before Moving On...
- ✅ Can I explain what a ratio is?
- ✅ Can I write a ratio in the correct order?
- ✅ Can I simplify ratios fully?
- ✅ Can I compare quantities using the same units?
- ✅ Can I share a quantity in a given ratio?
- ✅ Can I calculate a unit rate?
- ✅ Can I solve a direct-proportion problem?
- ✅ Can I use y = kx?
- ✅ Can I solve an inverse-proportion problem?
- ✅ Can I use y = k/x?
🎉 Well Done!
You can now compare quantities and solve problems using ratio, rate and proportion.
Check the order and units → Identify the relationship → Choose the correct method → Calculate → Check the result
📈 Your Progress
Lesson 14 of 19