Ratio and Proportions, Percentage and S.I. and C.I.

**Category : **7th Class

**Ratio and Proportions, Percentage and S.I. and C.I**.

**Ratio**

A ratio is a relation between two quantities of same kind, The ratio of a number x to another number y (where\[y\ne 0\]) is written as x : y.

- Example:

Daniel wants to divide ` 1530 between David and Michael in the ratio 8 : 9. Find the amount received by David.

(a) ` 720 (b) ` 810

(c) ` 900 (d) ` 820

(e) None of these

Ans. (a)

Explanation: Amount received by David \[=\,\,Rs.\,\,\frac{1530}{17}\times 8=Rs\,\,720\]

**Proportion**

A proportion is a name we give to a statement when two ratios are equal. It can be written in two ways:

- \[\frac{a}{b}=\frac{c}{d}\,\,(two\,\,equal\,\,fractions)\] \[a:b::c:d\,\,~\left( using\text{ }a\text{ }colon \right)\]

When two ratios are equal then, their cross products are equal.

That is, for the proportion, \[a\text{ }:\text{ }b=c:d,\text{ }a\text{ }\times \text{ }d==b\,\,\times \,\,c\]

In the proportion \[a:\text{ }b\text{ }::\text{ }c\text{ }:\text{ }d\], a and d are called extreme terms and b and c are called mean terms.

- Example:

If \[\mathbf{a}\text{ }:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{1}\text{ }:\text{ }\mathbf{5}\], find the ratio\[\mathbf{4a}\text{ }+\text{ }\mathbf{3b}\text{ }:\text{ }\mathbf{5a}\text{ }+\text{ }\mathbf{2b}\].

(a) 9 : 5 (b) 12 : 13

(c) 10 : 11 (d) 19 : 15

(e) None of these

Ans. (d)

- Example:

Jennifer mixes 600\[ml\] of orange juice with \[2l\] of apple juice to make a fruit drink. Find the ratio of orange juice to apple juice in its simplest from.

(a) 1 : 3 (b) 300 : 1

(c) 3 : 10 (d) 3 : 2

(e) None of these

Ans. (c)

Explanation: \[600:2000=\frac{600}{2000}=\frac{6\times 100}{20\times 100}=\frac{6}{20}=\frac{3\times 2}{2\times 10}=\frac{3}{10}=3:10\]

**Percentage**

Percentage is a fraction whose denominator is 100. The numerator of the such fraction is called the rate percent. For example 15 percent means \[\frac{15}{100}\] and denoted by 15 %.

- Example:

What percent of 2 km is 500 m?

(a) 25 % (b) 400 %

(c) 35 % (d) 45 %

(e) None of these

Ans. (a)

Explanation: Required percentage \[=\frac{500}{2\times 1000}\times 100=25%\]

**Increase and Decrease in Percentage**

To find the increase or decrease in percentage take the absolute value of the difference and divide it by the original value, then convert the resulting value into percent.

Note: Percentage increase and percentage decrease are measures of percent change, which is the extent to which something gains or loses.

- Example:

In the new budget, the price of LP.G increased by 10%. By how much percent a person should reduce his consumption so that his expenditure is not affected?

(a) 10% (b) \[9\frac{1}{11}%\]

(c) \[9\frac{1}{10}%\] (d) \[9\frac{1}{12}%\]

(e) None of these

Ans. (b)

Explanation: let original price of the L.P.G. be Rs. x

Then increased price \[=\,\,Rs.\,\,\frac{1}{100}\times \,\,x=\,\,\,Rs.\,\frac{x}{10}\]

New price of the L.P.G \[=\,\,Rs.\,\left( x+\frac{x}{10} \right)=\,\,Rs.\,\frac{11x}{10}\]

Reduction in consumption required \[=\left( \frac{\frac{x}{10}}{\frac{11x}{10}}\times 100 \right)=\frac{100}{11}%=9\frac{1}{11}%\]

**Profit and Loss**

In our day to day life we exchange things from others with money. During such transactions either we get profit or loss.

**Cost Price**

It is the price of an article at which a shopkeeper purchases the goods from manufacturer or wholesaler. In short it can be written as C.P.

**Selling Price**

It is the price of an article at which it is sold by the shopkeeper to the customer. In short it can be written as S.P.

**Profit and there is a Percent**

If S.P. > C.P., then there is a profit.

- Profit = S.P. ? C.P.
- Profit % = \[\text{Profit}\,%\,\,=\,\,\frac{\text{Profit}}{C.P.}\times 100\]

A mobile phone is sold for ` 576 at the loss of 4 %. What will be loss percent, if it is sold for ` 640?

(a) A gain of \[6\frac{2}{3}%\] (b) A loss of \[6\frac{2}{3}%\]

(c) A gain of \[3\frac{2}{3}%\] (d) A loss of \[3\frac{2}{3}%\]

(e) None of these

Ans. (a)

Explanation: Let C.P. be ` x, then loss \[\frac{4\times x}{100}=Rs.\frac{4\times }{100}\]

\[S.P=x-\frac{4x}{100}=576\,\,\Rightarrow \,\,\,\frac{96x}{100}=576\Rightarrow \,\,x=\,\,Rs.\,\,600.\]

Required gain% \[=\frac{40}{600}\times 100=\frac{20}{3}%=6\frac{2}{3}%\]

- Example:

The price of an article is increased from ` 5,600 to ` 6,400 in a particular day during winter season. Find the percentage increase.

(a) 14.29 % (b) 14.32%

(c) 14.23% (d) 12.56%

(e) None of these

Ans. (a)

Explanation: Increase in price = ` 6400 ? ` 5,600 = ` 800

Percentage increase \[=\frac{Increased\,\,price\,\times 100}{Original\,price}\]

\[=\,\,\,\frac{800\times 100}{5600}=\text{ }14.2857\text{ }=\text{ }14.29%\]

**Discount**

In our daily life whenever we go to the market, we see banners and big hoardings indicating discount or sale up to 50 % off or buy one get one free. These are different tacts to attract the customers to the market. Shopkeepers want to get maximum price for their goods and the customers are willing to pay as less as possible. Shopkeepers also offer different types of rebates in order to increase the sales or to finish old or damaged stock. This type of rebate on the price of an article is called discount.

** **

**Simple and Compound Interest**

In our daily life the transaction of money is a common phenomenon in business here transaction involves large amount of money. Money is borrowed from bank or from individuals for certain duration of time and at certain rate of interest. The sum further returned to the specified person or bank including the interest on the original sum. Interest is that excess money paid on borrowed amount

**Simple Interest**

If principle is P, rate of interest is R % per annum and time is T then simple interest will be:

\[S.I.=\,\,\frac{P\times R\times T}{100}\]

- Example:

Find the simple interest on ` 5,000 for 2 years at 8 % per annum.

(a) ` 800 (b) ` 900

(c) ` 850 (d) ` 875

(e) None of these

Ans. (a)

Explanation: Simple interest = ` \[\frac{5000\times 2\times 8}{100}=Rs.\,\,800\]

**Compound Interest**

If principle is P, rate of interest is R % per annum compounded annually and time is T then the compound interest will be:

\[C.I.=\,\,P{{\left( 1+\frac{R}{100} \right)}^{T}}-P\]

- Example:

Find the compound interest on Rs. 2,000 for 2 years at 10% per annum compounded annually.

(a) Rs. 350 (b) Rs. 2450

(c) Rs. 2350 (d) Rs. 420

(e) None of these

Ans. (d)

Explanation: \[C.I.\,\,=2000{{\left( 1+\frac{10}{100} \right)}^{2}}-2000=2000\times \frac{11}{10}\times \frac{11}{10}\,\,-2000\]

\[=\,\,\,Rs.(2420-2000)=\,\,Rs.\,\,420\]

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