**What is a Ratio?**

A

We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.

**ratio**is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.

A ratio is the comparison or
simplified form of two quantities of the same kind. This relation indicates how
many times one quantity is equal to the other. In other words, ratio is a
number, which expresses one quantity as a fraction of the other.

For example, ratio of 3 to 4
is 3 : 4.

The numbers forming the
ratio are called terms. The numerator, “3” , in this case, is known as the antecedent
and the denominator, “4” ,
in this case, is known as the consequent.

##
**Equivalent Ratios**

Let us divide a pizza into 8
equal parts and share it between Shreya and Surabhi in the ratio 2:6. The ratio
2:6 can be written as 2/6.

2/6 = 1/3. We know that 2/6
and 1/3 are called equivalent fractions. Similarly, we call the ratios 2:6 and
1:3 as equivalent ratios.

From a given ratio x : y, we can get equivalent ratios by multiplying the terms ‘x’ and ‘ y ‘by the same non-zero number.

For example,

1 : 3 = 2 : 6 = 3 : 9

4 : 5 = 12 : 15 = 16 : 20

From a given ratio x : y, we can get equivalent ratios by multiplying the terms ‘x’ and ‘ y ‘by the same non-zero number.

For example,

1 : 3 = 2 : 6 = 3 : 9

4 : 5 = 12 : 15 = 16 : 20

##
**Comparing Ratios**

To compare ratios, write them as fractions. The ratios are equal
if they are equal when written as fractions.

**Example:**

Are the ratios 3 to 4 and 6:8 equal?

The ratios are equal if 3/4 = 6/8.

These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal to 24, the answer is yes, the ratios are equal.

A ratio of 1:7 is not the same as a ratio of 7:1.

The ratios are equal if 3/4 = 6/8.

These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal to 24, the answer is yes, the ratios are equal.

A ratio of 1:7 is not the same as a ratio of 7:1.

##
**Proportion**

**Proportion**is represented by the symbol '=' or '::'

If the ratio a : b is equal to the ratio c : d, then a, b, c, d are said to be in proportion.

Using symbols, we write a : b = c : d or a : b :: c : d

When 4 terms are in
proportion, then the product of the two extreme terms (i.e. the first and the
fourth terms) should be equal to the product of two middle terms (i.e. the second
and the third terms)

A

For example, 3/4 = 6/8 is a proportion.

**proportion**is an equation with a ratio on each side. It is a statement that two ratios are equal.For example, 3/4 = 6/8 is a proportion.

When one of the four numbers in a proportion is unknown, cross
products may be used to find the unknown number. This is called solving the
proportion. Question marks or letters are frequently used in place of the
unknown number.

**Example 1:**Solve for n: 1 : 2 =

*n*: 4.

**Solution:**We can write the ratios in fraction form as 1/2 =

*n*/4

Using cross products we see that 2 ×

*n*= 1 × 4 = 4, so 2 ×

*n*= 4.

Dividing both sides by 2,

*n*= 4 ÷ 2 so that

*n*= 2.

**Example 2:**Prove that 16 : 12 and 4 : 3 are in proportion.

**Solution:**The product of the means = 12 × 4 = 48. The product of the extremes = 16 × 3 = 48

As Product of Means = Product of Extremes ∴ 16 : 12 and 4 : 3 are in proportion.

**Example 3:**Find the value of

*a*in 3 : 4 = 12 :

*a*

**Solution**

**:**We know that, Product of means = Product of extremes.

Therefore, 3 ×

*a*= 4 × 12; By dividing both sides by 3, we get

*a*= (4 × 12)/3 = 16

**Example
****4****:**** **A
van requires 2 litres of petrol to cover 48 km. Will 10 litres of petrol be
sufficient to cover 240 km?

**Solution:
**Let *x *km be the distance that the van covers using 10
litres of petrol. Then 2 : 48 :: 10 : *x*.

Then, 2*x* =
48 ×
10

Or, *x* = 240 km

Thus, 10 litres
of petrol is sufficient to cover 240 km.

**Example
****5****:**** **The
cost of 6 boxes of mangoes is ₹ 810. Will ₹ 3200 be enough to buy 25 such boxes
of mangoes?

**Solution:** Let *x *be the cost of 25
boxes of mangoes; then 6 : 810 :: 25 : *x*.

6*x* = 810 ×
25

Or, *x* = ₹ 3375

Therefore, we can
say that ₹ 3200 is not enough to buy 25 boxes of mangoes.

##
**Mean Proportional**

Mean proportional between

*a*and*b*is √*ab*.**Example 6:**Find the mean proportional of the numbers 10 and 1000.

**Solution**

**:**Mean proportional between a and b is √ab. Let the mean proportional of 10 and 1000 be x.

So, x = √10×1000 = √10000 = 100.

##
**Fourth Proportional**

**If a : b = c : d, then d is called the fourth proportional to a, b, c.**

**Example 7:**Find the fourth proportional of the numbers 12, 48, 16.

**Solution**

**:**Let fourth proportional is a. Now as per the concept above the product of extremes should be equal to the product of the means → 12/48 = 16/a → a = 64.

##
**Third Proportional**

a : b = c : d, then c is
called the third proportion to a and b.

**Example 8:**If 2, 5, y, 30 are in proportion, find the third proportional “y”.

**Solution**

**:**Here y is the third proportional. According to the concept 2/5 = y/30 → y = 12.

##
**Continued Proportion**

a, b and c are in continued proportion if a : b = b
: c. Here b is called the mean proportional and
is equal to the square root of the product of a and c.

Thus, b

Thus, b

^{2}= a × c → b = √ac
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