Ratio & Proportion — Grade VIII Mathematics

Chapter 4

Unit III · अङ्कगणित (Arithmetic)

Ratio & Proportion

अनुपात र समानुपात

📚 Grade VIII

⏱ TH 45 hrs (Unit)

🎯 3 Marks in Exam

🔢 K · U · A · HA

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Learning Outcomes

What you will achieve after studying this chapter

LO1

Compare quantities using ratio — Express the relationship between two quantities using ratio notation $a:b$ and reduce ratios to their simplest form by dividing by the HCF.

LO2

Identify and solve proportions — Determine whether two ratios form a proportion, and use cross-multiplication to find missing values in direct and inverse proportion problems.

LO3

Apply proportional reasoning — Solve real-life problems involving sharing quantities in a given ratio, scaling recipes, map reading, and speed–distance–time.

Learning Content

Ratio — A ratio compares two quantities of the same kind. Written as $a : b$ or $\dfrac{a}{b}$. Always simplify to lowest terms. E.g. $24:36 = 2:3$.

Proportion — Equality of two ratios: $a:b = c:d$ means $a \times d = b \times c$. The values $a,d$ are extremes; $b,c$ are means. Direct proportion: $\frac{x}{y}=k$. Inverse: $xy=k$.

Dividing in a ratio — To share an amount $T$ in ratio $m:n$, each share is $\dfrac{m}{m+n} \times T$ and $\dfrac{n}{m+n} \times T$ respectively.

Interactive Graph

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Key Formulas

Memorise these — they appear directly in SEE

Ratio

$$a : b = \frac{a}{b}$$

Both quantities must be in the same unit before forming a ratio.

Proportion — Product Rule

$$a \times d = b \times c$$

Product of extremes = Product of means

Direct Proportion

$$\frac{x_1}{y_1} = \frac{x_2}{y_2} = k$$

$k$ is the constant of proportionality; graph is a straight line through origin.

Inverse Proportion

$$x_1\,y_1 = x_2\,y_2$$

Product of corresponding values is always constant.

Compound Ratio

$$a:b \;\&\; c:d \;\Rightarrow\; ac:bd$$

Multiply antecedents together and consequents together.

Mean Proportional

$$a:b = b:c \;\Rightarrow\; b = \sqrt{ac}$$

$b$ is the geometric mean (mean proportional) of $a$ and $c$.

Dividing in Ratio

$$\text{Share}_1 = \frac{m}{m+n} \times T$$

Use for sharing money, time, or any quantity in ratio $m:n$.

Continued Proportion

$$\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \cdots$$

Each term is the geometric mean of the two terms surrounding it.

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Key Concepts

Deep understanding — beyond memorising formulas

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What is a Ratio?

A ratio compares two like quantities using division. If a bag has 12 red and 8 blue beads, ratio of red to blue is $12:8 = 3:2$. Always reduce using HCF.

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Testing Proportion

Check $2:3 = 4:6$ → extremes: $2\times6=12$, means: $3\times4=12$ ✓. If products match, it's a proportion.

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Direct Proportion

More workers → more bricks laid in the same time. Both quantities increase/decrease together. Graph: straight line through origin with slope $k$.

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Inverse Proportion

More workers → fewer days to finish a job. As one quantity increases the other decreases so that $x \times y = \text{constant}$.

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Splitting in a Ratio

Share Rs 700 in ratio $3:4$. Total parts = 7. First share: $\frac{3}{7}\times700 = \text{Rs }300$. Second: $\frac{4}{7}\times700 = \text{Rs }400$.