**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.,

\(\frac{\text{Weight of Seema}}{\text{Weight of Reema}}=\frac{\text{50 kg}}{\text{60 kg}}\)

\(=\frac { 6 }{ 5 }\)

So, the weight of Seema is \(\frac { 6 }{ 5 }\) 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:** \(\frac { 5 }{ 8 }\) = 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:**

\(\frac{12}{36}=\frac{1}{3}=1:3\) - 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

\(\frac{200}{2000}=\frac{1}{10}=1:10\) - 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 = \(\frac { 5 }{ 6 }\)

**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: 9.

**Solution:**

Express the given ratios as fraction

5 : 6 = \(\frac { 5 }{ 6 } \) and 7 : 9 = \(\frac { 7 }{ 9 } \)

Now find the L.C.M (least common multiple) of 6 and 9

The L.C.M (least common multiple) of 6 and 9 is 18.

Making the denominator of each fraction equal to 18, we have

\(\frac { 5 }{ 6 } \) = (5 ×3)/(6 ×3) = \(\frac { 15 }{ 18 } \) and \(\frac { 7 }{ 9 } \) = (7 ×2)/(9 ×2) = \(\frac { 14 }{ 18 } \)

Clearly, 15 > 14

Now, \(\frac { 15 }{ 18 } \) > \(\frac { 14 }{ 18 } \)

Therefore, 5 : 6 > 7 : 9.

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

**Solution:**

Find the GCD (or HCF) of numerator and denominator

GCD of 275 and 125 is 25

Divide both the numerator and denominator by the GCD

\(\frac{275 \div 25}{125 \div 25}\)

Reduced fraction:

\(\frac { 11 }{ 5 } \)

Therefore, \(\frac { 275 }{ 125 } \) simplified to lowest terms is \(\frac { 11 }{ 5 } \).

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

1 : 3, 5 : 12, 4 : 15 and 2 : 3

**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.

⇒ 2x + 3x = 1500

⇒ 5x = 1500

⇒ x = \(\frac { 1500 }{ 5 } \)

⇒ x = 300

∴ first son get 2x = 2 × 300 = 600

second son get 3x = 3 × 300 = 900

**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 rest are 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

(a) no. of blue pencil = 100 – 60 = 40

no. of total pencil = 100

Ratio = \(\frac { 40 }{ 100 } \) =\(\frac { 4 }{ 10 } \) =\(\frac { 2 }{ 5 } \)

(b) no. of red pencil = 60

no. of blue pencil = 40

Ratio = \(\frac { 60 }{ 40 } \) =\(\frac { 6 }{ 4 } \) =\(\frac { 3 }{ 2 } \)

(c) no. of red pencil = 60

no. of total pencil = 100

Ratio = \(\frac { 60 }{ 100 } \) =\(\frac { 6 }{ 10 } \) =\(\frac { 3 }{ 5 } \)

**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.

\(\frac { x }{ 12 } \) = \(\frac { 3 }{ 4 } \)

Now, after cross multiplying (the denominator of L.H.S. gets multiplied to the numerator of R.H.S. and similarly the denominator of R.H.S. with the numerator of L.H.S.), we get

4 × x = 3 × 12

4 × x = 36

x = \(\frac { 36 }{ 4 } \)

x = 9

So, the answer is x = 9.

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

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

10:20 = 20:x

\(\frac { 10 }{ 20 } \) = \(\frac { 20 }{ x } \)

20 × 20 = 10 × x

x = 40

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

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

The numbers is the continued proportion are in the form of \(\frac { a }{ b } \) = \(\frac { b }{ c } \)

Here, a = 14, b = 42 and c = x

Cross multiply

14 × x = 42 × 42

14 × x = 1764

∴ x = 126

∴ The value of x is 126.

**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.

cost of 12 bananas =₹24

cost of 1 banana = \(\frac { 24 }{ 12 } \) = ₹2

cost of 50 bananas = (50 ×2) = ₹100

**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.

12 : 10

48 : x

⇒ 12 × x = 48 × 10

x = 40

therefore it will take Rajesh 40min to travel 48km.