**What is a Ratio and Proportion**

**RATIO**

In our day-to-day life, we compare one quantity with another quantity of the same kind by using the ‘method of subtraction’ and ‘method of division’.

**Example:** The height of Seema is 1 m 67 cm and that of Reema is 1 m 62 cm. The difference in their heights is:

167 cm – 162 cm = 5 cm

Thus, we say Seema is 5 cm taller than Reema.

Similarly, suppose the weight of Seema is 60 kg and the weight of Reema is 50 kg. We can compare their weights by division, i.e.,

[latex s=2]\frac{\text{Weight of Seema}}{\text{Weight of Reema}}=\frac{\text{50 kg}}{\text{60 kg}}[/latex]

[latex s=2]=\frac { 6 }{ 5 }[/latex]

So, the weight of Seema is [latex]\frac { 6 }{ 5 }[/latex] times the weight of Reema.

When we compare two similar quantities by division, the comparison is called the** ‘ratio’**. It is denoted by **‘:’** and read as **‘is to’**.

**Example:** [latex s=2]\frac { 5 }{ 8 }[/latex] = 5 : 8 (read as 5 is to 8).

As shown in the above example a ratio is like a fraction or comparison of two numbers, where a numerator and a denominator is separated by a colon (:). The first term or the quantity (5), called antecedent means ‘that precedes’ and the second term, called consequent means ‘that follows’.

**Read More:**

Ratio and Proportion Rs Aggarwal Class 7 Solutions

Ratio and Proportion Rs Aggarwal Class 6 Solutions

**Properties of Ratio**

When we compare two quantities, the following points must be taken care of:

- A ratio is usually expressed in its simplest form.

**Example:**

[latex s=2]\frac{12}{36}=\frac{1}{3}=1:3[/latex] - Both the quantities should be in the same unit. So, ratio is a number with no unit involved in it.

**Example:**200 g : 2 kg

= 200 g : 2000 g

[latex s=2]\frac{200}{2000}=\frac{1}{10}=1:10[/latex] - The order of the quantities of a ratio is very important.

**Example:**5 : 6 is different from 6 : 5.

They are not equal.

5 : 6 ≠ 6 : 5

**Equivalent Ratios**

A ratio is similar to a fraction. So, if we divide or multiply the numerator (antecedent) and denominator (consequent) by the same number, we get an equivalent fraction (ratio).

**Example:** 5 : 6 = [latex s=2]\frac { 5 }{ 6 }[/latex]

**Comparison of Ratios**

To compare two ratios, we have to follow these steps:

**Step 1:** Convert each ratio into a fraction in its simplest form.

**Step 2:** Find the LCM of denominators of the fractions obtained in step 1.

**Step 3:** Convert the denominators equal to LCM obtained in step 2 in each fraction.

**Step 4:** Now, compare the numerators of the fractions; the fraction with a greater numerator will be greater than the other.

**Example 1:** Compare the ratio 5: 6 and 7: 8.

**Solution:** Here, 5 : 6 = 5/6 and 7 : 8 = 7/8.

**Example 2:** Convert the ratio 125 : 275 in its simplest form.

**Solution:**

**Example 3:** Write the following ratios in descending order:

4 : 3, 4 : 7, 7 : 10

**Solution:** We have,

**Example 4:** Mr Lai divides a sum of Rs. 1500 between his two sons in the ratio 2 : 3. How much money does each son get?

**Solution:** Let the first son get 2x and the second son get 3x.

**Example 5:** Two numbers are in the ratio 3 : 5 and their sum is 96. Find the numbers.

**Solution:** Let the first number be 3x and the second number be 5x.

Then, their sum = 3x + 5x = 96

8x = 96

x = 12

The first number = 3x = 3 × 12 = 36

The second number 5x = 5 × 12 = 60

**Example 6:** In a pencil box there are 100 pencils. Out of which 60 are red pencils and the restare blue pencils. Find the ratio of:

(a) blue pencils to the total number of pencils.

(b) red pencils to the total number of pencils.

(c) red pencils to blue pencils.

**Solution:** Total number of pencils in the pencil box = 100

Number of red pencils = 60

∴ Number of blue pencils = 100 – 60 = 40

**PROPORTION**

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. When two ratios are equal then such type of equality of ratios is called proportion and their terms are said to be in proportion.

**Example:** If the cost of 3 pens is Rs. 21, and that of 6 pens is Rs. 42, then the ratio of pens is 3 : 6, and the ratio of their costs is 21 : 42. Thus, 3 : 6 = 21:42. Therefore, the terms 3,6,21, and 42 are in proportion.

Generally, the four terms, a, b, c, and d are in proportion if a : b = c : d.

Thus, a : b : : c : d means a/b = c/d or ad = ad = bc

Conversely, if ad = be, then a/b = c/d or a : b : : c : d

Here, a is the first term, b is the second term, c is the third term, and d is the fourth term. The first and the fourth terms are called extreme terms or extremes and the second and third terms are called middle terms or means.

**Continued proportion**

In a proportion, if the second and third terms are equal then the proportion is called continued proportion.

Example: If 2 : 4 : : 4 : 8, then we say that 2, 4, 8 are in continued proportion.

**Mean proportion**

If the terms a, b, and c are in continued proportion, then ‘b’ is called the mean proportion of a and c.

**Example:** If a, b, c are in continued proportion, then

Mean proportion = b^{2} = ac

**Third proportion**

If the terms a, b, c are in continued proportion, then c is called the third proportion.

**Example 1:** Find x, where x : 3 : : 4 : 12.

**Solution:** Here, x, 3, 4, and 12 are in proportion.

**Example 2:** Find the third proportion of 10 and 20.

**Solution:** If a, b, c are in proportion, then b^{2} = ac.

**Example 3:** Find the value of x, if 14, 42, x are in continued proportion.

**Solution:** Here 14, 42, and x are in proportion.

**Example 4:** The cost of 1 dozen bananas is Rs. 24. How much do 50 bananas cost?

**Solution:** Let the cost of 50 bananas be x.

**Example 5:** Rajesh drives his car at a constant speed of 12 km per 10 minutes. How long will he take to cover 48 km?

**Solution:** Let Rajesh take x mins, to cover 48 km.

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