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Multiplying Fractions

There are 3 simple steps to multiply fractions

- Multiply the top numbers (the numerators).
- Multiply the bottom numbers (the denominators).
- Simplify the fraction if needed.

example: 1 / 2 × 2 / 5 =?

Step 1 Multiply the top numbers: 1 / 2 × 2 / 5 = 1 × 2 / = 2

Step 2 Multiply the bottom numbers: 1 / 2 × 2 / 5 = 1 × 2 / 2 × 5 = 2 / 10

Step 3 Simplify the fraction: 2 /10 = 1 / 5

Indics and surds

An index (plural: indices) is the power, or exponent, of a number. For example, a³ has an index of 3.

A surd is an irrational number that can be expressed with roots, such as ?2 5?19.

Example : (17)^{3.5} x (17)^{?} = 17^{8}

Let (17)^{3.5} x (17)^{x} = 17^{8}.

Then, (17)^{3.5 + }^{x} = 17^{8}

3.5 + x = 8

x = (8 – 3.5)

x = 4.5

Profit and loss

Cost price and selling price

Cost price (CP) is the price at which an article is purchased.

Selling price (SP) is the price at which an article is sold.

If selling price is more than cost price, profit(gain) occurs. If selling price is less than cost price, loss occurs.

In case of profit.

profit = selling price – cost price

selling price = cost price + profit

cost price = selling price – profit

In case of loss,

loss = cost price – selling price

selling price = cost price – loss

cost price = selling price + loss

Profit percentage and loss percentage

Profit percentage and loss percentage are always calculated on cost price unless otherwise stated.

In case of profit,

Profit percentage=profit×100 / cost price

selling price=cost price+cost price×profit percentage / 100

= cost price(100+profit percentage) / 100

Ratio and proportion

Ratio

Ratios are the mathematical numbers used to compare two things which are similar to each other in terms of units. For example, we can compare length of a pencil to length of a pen likewise distance to a unit that denotes distance. We can’t compare two things that are not similar to each other. Similarly, we can’t compare the height of a person to the weight of another person.

As an illustration, suppose the weight of Kamal is 5okg and the weight of Hassan is 100kg. A ratio of Kamal’s weight to Hassan’s weight can be found out by dividing Kamal’s weight to Hassan’s weight and vice versa. The ratio between Kamal’s and Hassan’s weight is 50/100= 1:2.

Proportion

Ratios compare things similar to each other. Further, these ratios are compared with each other using proportions. The purpose of comparing ratios is to deduce whether two distributions are equal or not. It additionally helps us to find out the more suitable proportion. When two ratios are the same, they are said to be proportionate to each other. A proportionate relation is represented by ‘::’ or ‘=’ sign.

Let’s assume you and your friend go out to buy notebooks. You both buy a total of 8 notebooks, which amount to 200 Rs. Your friend pays 50 Rs. while you pay 150 Rs. Now while returning your friend suggests that both of you receive 4 notebooks each. On the other hand, since you paid more, you suggest that you must receive notebooks while your friend gets 2 notebooks.

To decide which of you is correct, we can determine whether ratios of money paid and notebook distribution are equal or not. Here, the ratio of money you paid to that your friend paid is 150/50 = 3:1. The ratio according to your friend’s distribution is 4/4 = 1:1. Whereas, the ratio according to your distribution is 6/2= 3:1. Since the ratio of money paid and ratio according to your distribution is proportionate, your distribution will be correct.

Example : Divide 90 Rs. in ratio 1:2 between Ram and Karan.

There are two parts, 1 and 2, the sum of which is 3 parts. Hence among the 3 parts, Karan gets 2 and Ram gets 1. Therefore for 90 Rs (considered equivalent to 3 parts here) –

- Karan’s share = 2/3 ×90 = 60 Rs.
- Ram’s share = 1/3 ×90 = 30 Rs.

Here we have added the ratios as parts and then divided the total parts according to the ratio. Thus the sum of ratios is assumed to be equal to the total sum of rupees that needs to be divided among Karan and Ram.

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