PERCENTAGE
1. INTRODUCTION
The word 'Per cent' means hundred. Thus, 19 per cent means. 19 parts out of 100 parts. This can also be written as 19/100.
Therfore, per cent is fraction whose denominator is 100, and the numerator of this fraction is called the Rate percent. So,
| So, | 19 | = 19 per cent. Here, 19 per cent is the rate. The sign for per cent is '%'. |
| 100 |
Fractional Equlvalents of Important Percentages
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| 100% = 1 | ||||||||||||||||||||||||||||||||||||
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Note: Similarity along the horizontal rows are to be observed for memorizing table.
2. FRACTION TO RATE PER CENT
2. FRACTION TO RATE PER CENT
| To convert (or express) any fraction | a | to rate per cent, multiply it by 100 and put a(%) sign,. |
| b |
Images..
| Ex: Express | 3 | in rate per cent |
| 4 |
| Required rate per cent = | 3 | × 100% = 75% |
| 4 |
3. RATE PER CENT TO FRACTION
To convert a rate per cent to a fraction, divide it by 100 and delete the % sign.
| Ex. 8% can be converted to a fraction as | 8 | . |
| 100 |
4. RATE PER CENT OF A NUMBER
Rate per cent of a number is the product of equivaalent fraction (of rate per cent) and the number.
| ⇒ P% of A = | ( | P | ) | × A |
| 100 |
Ex: To find out 25% of 500
Soln. Required value = 25% of 500
| = | ( | 25 | ) | × 500 → equivalent fraction for 25% |
| 100 |
= 125
Relation Among Rate Per cent, Number and Value
Let us consider a number, N
Then N is considered as the base over which value of different rate per cent are found out.
| 10% of N = | 10 | × N = | N | (value) |
| 100 | 10 |
| 25% of N = | 25 | × N = | N | (value) |
| 100 | 4 |
and so on.
Therefore, it is found that as the rate per cent changes, its related value for the same number will also change.
Conversely, different values stand for different rate per cents of the same number. As in the above
| example, | N | stands for 10% of N ; | N | stands for 25% of N and so on. |
| 10 | 4 |
In the above context, a very useful realtion is derived as:
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Ex: 9% of what number is 36?
Soln. Using the relation 1.
| The required number (base number) = | 36 |
| 9% |
| = | 36 | × 100 |
| 9 |
= 400
Note:36 is the value and its rate % of base number = 9%
Ex: If 30% of a number is 48, then what is 70% of the number?
Soln. Here, unitary method can be used to save the time.
30% → 48
| ⇒ 1% → | 48 |
| 30 |
| ⇒ 70% → | 48 | × 70 = 112 |
| 30 |
Hence, the required value is 112.
EXPRESSING A GIVEN QUANTITY AS PERCENTAGE OF ANOTHER GIVEN QUANTITY
Let one given quantity be x and another given quantity be y, It is often asked to find what percentage of y is x. Here both quantities (x and y) should be in same units. If not, they should be converted into the same unit.
Concept
The question requires us to express one given quantity ‘x’ as a percentage of another given quantity ‘y’.
Since y is the basis of comparison, so, y will be in the denominator. But x is to be converted as percentage of y, hence x will be in the numerator of the fraction. Now to convert the fraction to percentage, we will multiply it by 100. So, we get
Let one given quantity be x and another given quantity be y, It is often asked to find what percentage of y is x. Here both quantities (x and y) should be in same units. If not, they should be converted into the same unit.
Concept
The question requires us to express one given quantity ‘x’ as a percentage of another given quantity ‘y’.
Since y is the basis of comparison, so, y will be in the denominator. But x is to be converted as percentage of y, hence x will be in the numerator of the fraction. Now to convert the fraction to percentage, we will multiply it by 100. So, we get
| The required percentage = | x | × 100% |
| y |
Ex: To find '30 is what per cent of 150’ or ‘what percentage of 150 is 30?
Soln. Using the earlier concept, we find here that 150 is the basis of comparison and hence 150 will be in the denominator.
Soln. Using the earlier concept, we find here that 150 is the basis of comparison and hence 150 will be in the denominator.
| The required percentage = | 30 | × 100% |
| 150 |
= 20%.
Ex: ?% of 320 = 86.4
soln. Here, 320 is the basis of comparison and it will be in the denominator.
| ∴ required percentage = | 86.4 | × 100% |
| 320 |
= 27%
CONVERTING A PERCENTAGE INTO DECIMALS
Case I
Let the percentage be a positive integer, then
Place a decimal point after two places from the extreme right of the integer to convert it into a decimal if the percentage is a single digit number, add one zero to the left of it and then place the decimal point for its conversion. % sign is removed after conversion.
Case I
Let the percentage be a positive integer, then
Place a decimal point after two places from the extreme right of the integer to convert it into a decimal if the percentage is a single digit number, add one zero to the left of it and then place the decimal point for its conversion. % sign is removed after conversion.
| Ex: 67% may be converted into decimals as 0.67. because 67% = | 67 | = 0.67 |
| 100 |
Case II
Let the percentage be a decimal fraction
The percentage being a decimal fraction, shift decimal by two places to the left. Add zero to the left of the fraction, if needed.
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Case III
Let the percentage is fraction
Let the percentage be a decimal fraction
The percentage being a decimal fraction, shift decimal by two places to the left. Add zero to the left of the fraction, if needed.
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Case III
Let the percentage is fraction
| If the percentage is a fraction of the form | a | , then convert it into a decimal fraction and then follow |
| b |
| Ex: | 1 | % is equivalent to 0.25% which may be converted into decimals as | |
| 4 |
CONVERTING A DECIMAL INTO A PERCENTAGE
In this case, the method of 6 is reversed, i.e. shift the decimal point two places to the right. Add zero to the extreme right if required. Then add % sign.
EFFECT OF PERCENTAGE CHANGE ON ANY QUANTITY (NUMBER)
If any number (quantity) is increased by x%, then
In this case, the method of 6 is reversed, i.e. shift the decimal point two places to the right. Add zero to the extreme right if required. Then add % sign.
EFFECT OF PERCENTAGE CHANGE ON ANY QUANTITY (NUMBER)
If any number (quantity) is increased by x%, then
| new number (quantity) = original number × | ( | 100 + x | ) |
| 100 |
or
= original number × (1 + decimal equivalent of x%).
Similarly, if any number (quantity) is decreased by x%, then
| new number (quantity) = original number × | ( | 100 - x | ) |
| 100 |
or
= original number × (1 - decimal equivalent of x%).
Note: In case of percentage decrease, a (-) ve sign is put before x, otherwise the formula is same.
Ex: The present salary of A is Rs 3000. This will be increased by 15% in the next year. What will be the increased salary of A?
Soln. Here, the salary is to be increased by 15%.
15% is equivalent to 0.15
| ∴ the increased salary = 3000 (1 + 0.15) or 3000 × | 100 + 15 |
| 100 |
= 3000 × 1.15
= Rs 3450.
9. TWO STEP CHANGE OF PERCENTAGE FOR A NUMBER
In the first step, a number is changed (increased of decreased) by x%, and in the second step, this changed number is again changed (increased or decreased) by y%, then net percentage change on the original number can be conveniently found out by using the following formula,
| net% change = x + y + | xy |
| 100 |
(+ or -)
If x or y indicates in percentage, then put a (-)ve sign x or y, otherwise positive sign remains.
Ex: If a number is increased by 12% and then decreased by 18%, then find the net percentage change in the number.
Soln. Using the formula (2)
Soln. Using the formula (2)
| net % change = x + y + | xy |
| 100 |
where x = 12 y = -18
| ⇒ net % change = 12 - 18 + | (12) × (-18) |
| 100 |
= -6 - 2.16
= -8.16
(-) sign signfies that there is percentage decrease in the result. Therefore -8.16 indicates net 8.16% decrease of the given number as a result of 12% increase and 18% decrease.
It also implies that 12% increase and 18% decrease are equivalent to 8.16% decrease.
PERCENTAGE CHANGE AND ITS EFFECT ON PRODUCT
If A is cahnged (increased or decreased) by x%, and also B is changed (increased or decreased) by y%. then the net percentage change (increase/decrease) of the product of A and B can be fount out easily by the formula
| net % change in product = x + y + | xy |
| 100 |
(+ or -)
If x or y indicates decrease in percentage, then put a (-)ve sign before x or y, otherwise positive sign remains.
This formula (2A) is same as formula (2)
Above formula can be used to find out the net percentage change, if it involves the product of any two variable quantities which have also the % change.
This formula (2A) is same as formula (2)
Above formula can be used to find out the net percentage change, if it involves the product of any two variable quantities which have also the % change.
Application of the formula (2A)
This formula (2A) can be used to find out
(a) % effect on expenditure, when rate and consumption are changed, since rate × consumption = expenditure [A × B = product]
(b) % effect on area of rectangle/ square/ triangle/circle, when its sides/radius are changed, since
side1 × side2 = area, or radius × radius = area [A × B = product]
(c) % effect on distance covered, when time and speed are changed, since time × speed = distance.
[A × B = product]
Example: If the length of rectangle increase by 30% and the breadth decrease by 12%, then find the % change in the area of the rectangle.
Soln. Since, length × breadth = area, and both the length and breadth are changed, so using the formula (2A), we get
| net % change in product = x + y + | xy |
| 100 |
where x = 30, y = -12
| ⇒ net % change in area = 30 - 12 + | 30 × (-12) |
| 100 |
= 18 - 3.6
= + 14.4
It implies that there is 14.4% increase in the area of the rectnagle.
11. TO KEEP THE PRODUCT OF TWO VARIABLE QUANTITY AS FIXED
As we have in 11, A × B = product, where A and B are two quantities which are changing and product is also changing.
Now, we want to keep the product fixed, even if A and B are changed (increased/ decreased).
Then, if one quantity increases, the other quantity will decrease and vice-versa so that product remains unchanged.
Hence the net percentage effect on product is zero in the formula (2A).
Put net % change in product = 0 in formula (2A).
As we have in 11, A × B = product, where A and B are two quantities which are changing and product is also changing.
Now, we want to keep the product fixed, even if A and B are changed (increased/ decreased).
Then, if one quantity increases, the other quantity will decrease and vice-versa so that product remains unchanged.
Hence the net percentage effect on product is zero in the formula (2A).
Put net % change in product = 0 in formula (2A).
| x + y + | xy | = 0. |
| 100 |
| y = - | x | × 100 (-)ve sign shows decrease |
| 100 + x |
From the above derivation, we thus find that
| if one quantity A increases by x%, then other quantity B decrease by |
|
and if one quantity A decrease by x%, then putting (-)x, in place of x,
| we find that the other quantity B increase by |
|
(a) % change in either rate or consumption, when expenditure is to be kept fixed, because, rate × consumption = expenditure [A × B = product = fixed]
(b) % change in either length or breadth, when area of rectangle is to be kept fixed, because, length × breadth = area [A × B = product = fixed]
(c) % change un either time or speed, when distance is to be kept fixed because, time × speed = distance [A ×B = product = fixed]
Ex: If the price of Coffee is increased by 10%, then by how much percentage must a house wife reduce her consumption, to have no extra expenditure?
Soln. Since Price × consumption = expenditure and expenditure has to be kept fixed (or unchanged),so, when the price increase by 10%,
Soln. Since Price × consumption = expenditure and expenditure has to be kept fixed (or unchanged),so, when the price increase by 10%,
| the % reduction in consumption = | 10 | × 100% |
| 100 + 10 |
| = 9 | 1 | %. |
| 11 |
RATE CHANGE AND CHANGE IN QUANTITY AVAILABLE FOR FIXED EXPERNDITURE
Let the original rate of an item = Rs x per unit quantity
Expenditure is fixed
Let the original rate of an item = Rs x per unit quantity
Expenditure is fixed
| Quantity of the item available = | Expenditure |
| x |
Now, the original rat (x) changes (increase/decrease) to a new rate. since the amount of Expenditure is fixed, so, with the change in rate, it is obvious that the quantity of the item available in equation (a) will also change (decrease/increase) accordingly.
Hence, due to rate change,
Hence, due to rate change,
| New, quantity of the item availabe = | Expenditure |
| New rate |
| ⇒ original quantity available ± change in quantity available = | Expenditure | , Using equation (a), we get |
| New rate |
Ex: A reduction of 25% in the price of sugar enables the person to get 10 kg more on a purchase for Rs. 600. Find the reduce rate of sugar.
Soln. Let the original rate = Rs x per kg.
Since, there is a rate reduction of 25%, so,
New rate (or reduced rate) = (1 – 0.25) x
Soln. Let the original rate = Rs x per kg.
Since, there is a rate reduction of 25%, so,
New rate (or reduced rate) = (1 – 0.25) x
| = 0.75 x = | 3 | x |
| 4 |
Expenditure = Rs 600.
Using the formula (3)
| Expenditure | + change in quantity available = | Expenditure |
| x | New Rate |
| ⇒ | 600 | + 10 = | 600 | |
| x |
|
(+10, for quantity available increases after rate change)
| ⇒ | 600 | ( | 4 | - 1 | ) | = 10 ⇒ x = 20 |
| x | 3 |
| Therefore, reduced rate = | 3 | x = | 3 | × 20 = Rs 15/kg. |
| 4 | 4 |
12. % EXCESS OR % SHORTNESS
| When a number A exceeds the another number B by x %. then % short ness of B | x | × 100 |
| 100 + x |
| It implies that B is less than A by | x | × 100%. |
| 100 + x |
| Similarly, if a number A is short of (or less than) B by x%, then % excess of B = | x | × 100 |
| 100 - x |
| i.e. B is more than A by | x | × 100% |
| 100 - x |
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