Cbse Class 6 Maths Notes Chapter 12

Comparison by Taking Difference:

1. We often employ the approach of taking differences between quantities when comparing quantities of the same type.

2. In some cases, a comparison by difference is not preferable to a comparison by division.

3. When we examine the two quantities in terms of ‘how many times,' this comparison is called ‘Ratio’.

Comparison by Division:

1. In many cases, division is used to make a more meaningful comparison of amounts, i.e., seeing how many times one quantity is to the other quantity.

Comparison by ratio is the name given to this procedure.

2. The sign $$:$$ is used to represent a ratio.

3. The two quantities must be in the same unit to be compared via ratio. If they aren't in the same unit, they should be before the ratio is calculated.

4. By multiplying or dividing the numerator and denominator by the same number, we can derive equivalent ratios.

Example:

Isha weighs $$\text{10 kg}$$ whereas her father weighs $$\text{50 kg}$$. The weights of Isha's father and Isha are said to be in a $$\text{50:10 = 5:1}$$ ratio.

5. The same ratio can occur in a variety of circumstances.

6. It's worth noting that the $$\text{a : b}$$ ratio is not the same as $$\text{b : a}$$. As a result, the order in which quantities are taken to express their ratio is important.

For example, $$\text{5 : 3}$$ ratio is not the same as the $$\text{3 : 5}$$ ratio.

7. A ratio can be expressed as a fraction; for example, the ratio $$\text{7 : 9}$$ can be expressed as $${\displaystyle \frac{7}{9}}$$.

8. In its simplest form, a ratio can be represented.

For example, the ratio $$\text{78 : 39}$$ is considered as $${\displaystyle \frac{\text{78}}{39}}$$ .

In its simplest form, a ratio can be represented as $${\displaystyle \frac{\text{78}}{39}}={\displaystyle \frac{2}{1}}$$.

Hence, the lowest form of the ratio $$\text{78 : 39}$$ is $$\text{2 : 1}$$.

9. If the fractions corresponding to two ratios are the same, they are comparable.

Example: 

$$\text{1 : 2}$$ is identical to $$\text{5 : 10}$$ or $$\text{8 : 16}$$.

10. We say two ratios are in proportion if they are equal, and we use the symbols ‘$$::$$’ or ‘$$=$$' to equate them.

11. We say that two ratios are not in proportion if they are not equal.

The four quantities involved in a statement of proportion are known as respective terms when they are taken in order.

Extreme terms are the first and fourth terms. Whereas, Middle terms are the second and third terms.

Example:

(i) Are $$16,48,17$$ and $$51$$ in proportion?

Ans:

$\text{Ratio of }\text{ 16 to 48 = }{\displaystyle \frac{\text{16}}{\text{48}}}$

$\text{ = }{\displaystyle \frac{\text{1}}{\text{3}}}$

$\text{ = 1 : 3}$

$\text{Ratio of }\text{ 17 to 51 = }{\displaystyle \frac{\text{17}}{51}}$

$\text{ = }{\displaystyle \frac{\text{1}}{\text{3}}}$

$\text{ = 1 : 3}$

Since, $$16:48=17:51$$

Therefore, $$16,48,17$$ and $$51$$ are in proportion.

Here, $$16$$ and $$51$$ are the extreme terms whereas, $$48$$ and $$17$$ are the Middle terms.

(ii) Do the ratios $$\text{15 cm}$$ to $$\text{3 m}$$ and $$\text{24 seconds}$$ to $$\text{5 minutes}$$ form a proportion?

Ans:

$$\text{Ratio of }\text{ 15 cm to 3 m = 15 : 3}\times 100$$ 

Since, $$\left(\text{1 m = 100 cm}\right)$$,

Therefore, we get,

$\text{Ratio of }\text{ 15 cm to 3 m = 15 : }300$

$\text{Ratio of }\text{ 15 cm to 3 m = 1 : }20$

$$\text{Ratio of }\text{ 24 sec to 5 minutes = 24 : 8}\times 60$$ 

Since, $$\left(\text{1 minute = 60 seconds}\right)$$,

Therefore, we get,

$\text{Ratio of }\text{ 24 sec to 5 minutes = 24 : }300$

$\text{Ratio of }\text{ 24 sec to 5 minutes = 1 : }12.5$

Since, $$1:20\ne 1:12.5$$

Therefore, the given ratios $$\text{5 cm}$$ to $$\text{3 m}$$ and $$\text{24 seconds}$$ to $$\text{5 minutes}$$ do not form a proportion.

12. The unitary method is a method that involves first determining the value of one unit and then determining the value of the required number of units.

Example:

If the cost of a dozen soaps is $$\text{Rs}\text{. 250}$$ , what will be the cost of $$\text{23}$$ such soaps?

Ans:

We have, a

Cost of a dozen soaps is $$\text{Rs}\text{. 250}$$.

Since, $$\text{1 dozen = 12}$$

Therefore,

$\text{Cost of }\text{ 1 soap = }{\displaystyle \frac{\text{250}}{\text{12}}}$

$\text{ = Rs}\text{. 20}\text{.83}$

Therefore,

$\text{Cost of }\text{ 23 soaps = 20}\text{.83}\times 23$

$\text{ = Rs}\text{. 479}\text{.09}$

Hence, the cost of $$\text{23}$$ soaps is $$\text{Rs}\text{. 479}\text{.09}$$.

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