Ratio Proportion Percentage and S-I and C-I

**Category : **7th Class

**Ratio and Proportion 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 L of water between David and Michael in the ratio\[8:9\]. Find the quantity received by David.

(a) L 720 (b) L 810

(c) L 900 (d) L 820

(e) None of these

**Answer **(a)

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

**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)
- \[a:b::c:d\] (using a colon)

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

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

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

**Example:**

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

(a) \[9:5\] (b) \[12:13\]

(c) \[10:11\] (d) \[19:15\]

(e) None of these

**Answer **(d)

**Explanation: \[\frac{a}{b}=\frac{1}{5}\Rightarrow b=5a\]**

Now, \[\frac{4a+3b}{5a+2b}=\frac{4a+3\times 5a}{5a+2\times 5a}=\frac{4a+15a}{5a+10a}=\frac{19a}{15a}=19\,:\,15\]

**Example:**

Jennifer mixes 600 ml of orange juice with 2 I 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

**Answer **(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 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

**Answer** (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 L.P.G. increased by 10 %. By how much percent a person should reduce his consumption so that his expenditure is not affected?

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

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

(e) None of these

**Answer **(b)

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

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

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 Profit Percent**

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

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

** **

**Loss and Loss Percent**

If\[S.P.<C.P\]., then there is a loss.

\[\Rightarrow Loss=C.P.-S.P.\]\[\Rightarrow Loss%=\frac{Loss}{C.P.}\times 100\]

** **

**Example:**

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

**Answer** (a)

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

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

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

**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 where 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 Rs. 5,000 for 2 years at 8 % per annum.

(a) Rs. 800 (b) Rs. 900

(c) Rs. 850 (d) Rs. 875

(e) None of these

**Answer **(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

Answer (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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