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\[ 2:5 \]
Welcome to Grade 8 Ratio. A ratio compares quantities of the same kind. In this lesson you'll learn to simplify ratios, compare them, share a quantity in a given ratio, and increase or decrease a number in a given ratio.
\[ \text{concentrate}:\text{water} = 2:5 \]
A juice recipe uses 2 cups of concentrate and 5 cups of water. The ratio of concentrate to water is \(2:5\). A ratio is simply a way of comparing two (or more) quantities of the same kind — here, both quantities are measured in cups.
\[ 2:5 \;\neq\; 5:2 \]
What is the ratio of water to concentrate in the same recipe? It's \(5:2\) — the numbers swap because the order of the words swapped. \(2:5\) is not the same as \(5:2\). Teaching point: the order of the numbers must always match the order of the words. "Boys to girls" means boys first, girls second.
\[ 2+5=7 \text{ total parts} \]
Add the parts of a ratio together to find the total number of parts. In our juice recipe, \(2:5\) has \(2+5=7\) total parts. This idea of "total parts" is the key to sharing a quantity in a given ratio later in this lesson.
\[ a:b \qquad \text{or} \qquad a:b:c \]
Key words for this lesson:
| Term | Meaning |
|---|---|
| Ratio | A comparison of quantities of the same type, e.g. \(8:12\) shells |
| Simplest form | A ratio where the parts share no common factor except 1, e.g. \(8:12 = 2:3\) |
| HCF | The highest number that divides every part exactly |
| Sharing ratio | Dividing a total into unequal parts, e.g. share R200 in \(3:2\) |
| Increasing ratio | Making a number bigger according to a ratio |
| Decreasing ratio | Making a number smaller according to a ratio |
\[ a:b \;=\; (a \div \text{HCF}) : (b \div \text{HCF}) \]
A ratio is simplified the same way as a fraction: divide every part by the same number — usually the HCF (highest common factor) of all the parts.
\[ 18:24 = 3:4 \]
Simplify \(18:24\). The HCF of 18 and 24 is 6. \(18:24 = (18\div6):(24\div6) = 3:4\).
\[ 24:36:60 = 2:3:5 \]
The same method works for three-part ratios. Simplify \(24:36:60\): the HCF of 24, 36 and 60 is 12, so \(24:36:60 = 2:3:5\).
Important: you must divide every part by the same number.
❌ Incorrect: \(24:36:60 = 2:3:60\) (only two parts were divided)
✔ Correct: \(24:36:60 = 2:3:5\)
\[ 75\text{ m}:125\text{ cm} \;\rightarrow\; 7500\text{ cm}:125\text{ cm} \]
If a ratio compares different units, convert them to the same unit first — only then simplify.
\[ 75\text{ m}:125\text{ cm} = 60:1 \]
Simplify \(75\text{ m}:125\text{ cm}\). Convert metres to centimetres: \(75\text{ m} = 7500\text{ cm}\), giving \(7500:125\). Divide both parts by 125: \(7500\div125=60\) and \(125\div125=1\). So \(75\text{ m}:125\text{ cm} = 60:1\).
\[ 9\text{ min}:600\text{ s} = 9:10 \]
Simplify \(9\text{ min}:600\text{ s}\). Convert minutes to seconds: \(9\text{ min} = 9\times60 = 540\text{ s}\), giving \(540:600\). Divide both parts by 60: \(540:600 = 9:10\).
\[ a:b \;=\; (a \div \text{HCF}) : (b \div \text{HCF}) \]
To simplify a ratio: 1) convert to the same unit if needed, 2) find the HCF of all the parts, 3) divide every part by the HCF.
\[ a:b \;\rightarrow\; \dfrac{a}{b} \]
To compare two ratios, write each one as a fraction, then compare the fractions using cross multiplication.
\[ 9\times6=54 \qquad 25\times2=50 \]
Which ratio is greater: \(9:2\) or \(25:6\)? Write as fractions: \(\tfrac{9}{2}\) and \(\tfrac{25}{6}\). Cross multiply: \(9\times6=54\) and \(25\times2=50\). Since \(54>50\), \(9:2 > 25:6\).
\[ 9:2 \;>\; 25:6 \]
Common mistake: learners often think \(25:6\) is bigger because 25 is a bigger number than 9. That's incorrect — a ratio compares the relationship between its two numbers, not just the size of the first number. Always convert to fractions and cross multiply to be sure.
\[ \text{value of 1 part} = \text{total} \div \text{total parts} \]
Use this method every time you share a quantity in a ratio:
1. Add the ratio parts together.
2. Divide the total quantity by the number of parts.
3. Multiply each ratio part by the value of one part.
4. Check that the shares add back up to the total.
\[ R10\,000 : R6\,000 : R4\,000 \]
Share R20 000 among three brothers in the ratio \(5:3:2\). Total parts: \(5+3+2=10\). Value of one part: \(R20\,000 \div 10 = R2\,000\). Shares: \(5\times R2\,000 = R10\,000\), \(3\times R2\,000 = R6\,000\), \(2\times R2\,000 = R4\,000\).
Check: \(R10\,000 + R6\,000 + R4\,000 = R20\,000\) ✔
\[ \text{share} = \text{ratio part} \times \text{value of 1 part} \]
Sharing in a ratio always follows the same four steps: add the parts, divide the total by the number of parts, multiply each part by that value, then check your shares add back to the total.
\[ \text{new number} \;>\; \text{original number} \]
When a number is increased in a given ratio, the answer must always come out bigger than the original. The original number corresponds to the smaller part of the ratio, and the new number corresponds to the larger part.
\[ 60 \times \dfrac{5}{3} = 100 \]
Increase 60 in the ratio \(3:5\). Because the number increases, 60 corresponds to the smaller part, 3. One part: \(60 \div 3 = 20\). The new number corresponds to 5 parts: \(5\times20=100\). So 60 increased in the ratio \(3:5\) is 100.
Shortcut: \(60 \times \tfrac{5}{3} = 100\).
\[ \text{new} = \text{original} \times \dfrac{\text{larger part}}{\text{smaller part}} \]
The fast way to increase a number in a ratio \(a:b\) (where \(b>a\)): multiply the original number by \(\tfrac{b}{a}\). Always check your answer is bigger than you started with.
\[ \text{new number} \;<\; \text{original number} \]
When a number is decreased in a given ratio, the answer must always come out smaller than the original. The original number corresponds to the larger part of the ratio, and the new number corresponds to the smaller part.
\[ 120 \times \dfrac{4}{5} = 96 \]
Decrease 120 in the ratio \(5:4\). Because the number decreases, 120 corresponds to the larger part, 5. One part: \(120 \div 5 = 24\). The new number corresponds to 4 parts: \(4\times24=96\). So 120 decreased in the ratio \(5:4\) is 96.
Shortcut: \(120 \times \tfrac{4}{5} = 96\).
\[ \text{new} = \text{original} \times \dfrac{\text{smaller part}}{\text{larger part}} \]
The fast way to decrease a number in a ratio \(a:b\) (where \(a>b\)): multiply the original number by \(\tfrac{b}{a}\). Always check your answer is smaller than you started with.
\[ 3:2 \;\neq\; \dfrac{3}{2} \]
| Misconception | Correction |
|---|---|
| Reversing the order of the ratio. | Always match the words: "boys to girls" means boys first, girls second. |
| Simplifying without converting units first. | Convert to the same unit before you simplify. |
| Dividing only one part of a ratio. | Whatever you do to one part, do to every part. |
| Comparing ratios by looking only at the first number. | Use fractions and cross multiplication instead. |
\[ \text{increase} \Rightarrow \text{bigger} \qquad \text{decrease} \Rightarrow \text{smaller} \]
| Misconception | Correction |
|---|---|
| Confusing a sharing ratio with a fraction. | In \(3:2\), the total parts are \(3+2=5\). The first share is \(\tfrac{3}{5}\) of the total, not \(\tfrac{3}{2}\). |
| Increasing a number but getting a smaller answer. | For an increase, the answer must always be bigger than the original. |
| Decreasing a number but getting a bigger answer. | For a decrease, the answer must always be smaller than the original. |
\[ \begin{aligned} \text{Simplify:}\;&a:b=(a\div HCF):(b\div HCF) \\ \text{Compare:}\;&\text{cross multiply the fractions} \\ \text{Share:}\;&\text{part}\times(\text{total}\div\text{total parts}) \\ \text{Increase/Decrease:}\;&\text{original}\times\dfrac{\text{new part}}{\text{old part}} \end{aligned} \]
You now know how to:
• Simplify ratios, including ones with different units
• Compare ratios using fractions and cross multiplication
• Share a quantity according to a given ratio
• Increase a number in a given ratio
• Decrease a number in a given ratio
• Spot and avoid the most common ratio mistakes
Ready to test yourself? Your teacher will share the quiz for this topic.