Indicies - Explanation, Examples and Laws of Indicies

 

                           Ratio and Proportion

Introduction

Ratios and proportions are some of the most practical concepts in mathematics. They help us describe relationships between quantities and maintain balance or fairness in everyday situations.

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1. What Is a Ratio?

A ratio is a way of comparing two or more quantities of the same kind.
If quantity $a$ is compared to quantity $b$, their ratio is written as:

$$a : b \quad \text{or} \quad \frac{a}{b}$$

For example:

• The ratio of 10 apples to 5 oranges is $10 : 5 = 2 : 1$
  

• The ratio of 12 boys to 18 girls is $12 : 18 = 2 : 3$
  

Important: Ratios compare quantities in the same unit. You cannot compare 3 meters to 4 kilograms.

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Simplifying Ratios

To simplify a ratio, divide both terms by their highest common factor (HCF).

Example 1:
Simplify $20 : 15$

$$\text{HCF of 20 and 15 is 5}$$
$$20:15=4:3$$

Answer: $4 : 3$

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Example 2:

If there are 25 boys and 30 girls in a class, find the ratio of boys to girls.

$$25 : 30 = \frac{25}{5} : \frac{30}{5} = 5 : 6$$

Answer: $5 : 6$

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2. Equivalent Ratios

Two ratios are said to be equivalent if they represent the same comparison when simplified.

Example 3:
Are $2 : 3$ and $8 : 12$ equivalent?

$$2 : 3 = \frac{2}{3}, \quad 8 : 12 = \frac{2}{3}$$

Answer: Yes, they are equivalent.

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3. Dividing a Quantity in a Given Ratio

When dividing an amount in a ratio, first find the total number of parts, then share the quantity according to each part’s value.

Example 4:
Divide 350 in the ratio $2 : 5$

$$\text{Total parts} = 2 + 5 = 7$$
$$\text{1st part} = \frac{2}{7} \times 350 = 100$$
$$\text{2nd part} = \frac{5}{7} \times 350 = 250$$

Answer: $100$ and $250$

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4. What Is a Proportion?

A proportion is an equality between two ratios.
If:

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

then $a, b, c,$ and $d$ are said to be in proportion, written as:

$$a : b = c : d$$

In any proportion:

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

This is known as the cross multiplication rule.

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Example 5:

Check if $3, 4, 9, 12$ are in proportion.

$$3:4=\frac{3}{4}=0.75,9:12=\frac{9}{12}=0.75$$

Answer: They are in proportion.

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Example 6:

Are $5, 10, 15, 20$ in proportion?

$$5 : 10 = \frac{1}{2}, \quad 15 : 20 = \frac{3}{4}$$

Answer: They are not in proportion.

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5. Finding the Missing Term in a Proportion

If three terms of a proportion are known, the fourth can be found using:

$$a : b = c : d \Rightarrow d = \frac{b \times c}{a}$$

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Example 7:

Find $x$ if $3 : 4 = 9 : x$

$$3x = 4 \times 9$$
$$x = \frac{36}{3} = 12$$

Answer: $x=12$

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Example 8:

Find $y$ if $8 : y = 6 : 9$

$$8 \times 9 = 6y$$
$$72 = 6y$$
$$y=12$$

Answer: $y = 12$

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6. Word Problems on Ratio and Proportion

Example 9:

The ratio of red to blue pens is 2:5. If there are 70 pens in total, how many are red and how many are blue?

$$\text{Total parts} = 2 + 5 = 7$$$$\text{Red pens} = \frac{2}{7} \times 70 = 20$$
$$\text{Blue pens}=\frac{5}{7}\times 70=50$$

Answer: 20 red pens and 50 blue pens

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Example 10:

If a car travels 180 km with 12 liters of petrol, how far will it go with 15 liters?

$$\frac{180}{12} = \frac{x}{15}$$ $$12x = 180 \times 15$$
$$x=\frac{2700}{12}=225$$

Answer: The car will travel 225 km.

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Summary

• A ratio compares two or more quantities.

• A proportion shows that two ratios are equal.

• The cross multiplication rule is used to find missing terms.

• Ratios and proportions are widely applied in daily life — from sharing profits to calculating speed and scale.

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Assignment

Try these to test your understanding 

1. Simplify $42:56$

2. Find the ratio of 2 meters to 50 centimeters.

3. Divide 270 in the ratio $3 : 6$
   

4. Are $2, 3, 4, 6$ in proportion?

5. Find $x$ if $5 : 8 = 10 : x$
   

6. Find $y$ if $7 : 14 = y : 20$
   

7. The ratio of boys to girls in a class is $4 : 5$. If there are 81 students in total, find the number of boys and girls.

8. A sum of money is shared between Ama and Kofi in the ratio 3:2. If Ama gets ₵90, how much does Kofi get?

9. A recipe uses sugar and flour in the ratio $1 : 4$. If 12 cups of flour are used, how much sugar is needed?

10. The ratio of the ages of a father and his son is $7 : 3$. If the sum of their ages is 60 years, find their ages.

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Conclusion

Ratios and proportions may look simple, but they form the backbone of percentages, scale drawings, and financial calculations. Try to solve more examples everyday on your own because consistency is key to making math simpler. See you in the next topic. Byeeeeeee👋