Maths formulas # 03 :)

Share This:

Share Tweet Pin

RATIO AND PROPORTION

• Ratio:

The ratio of two quantities a and b in the same units, is the fraction   and we write it as a : b.In the ratio a : b, we call a as the first term or antecedent and b, the second term orconsequent.

Eg. The ratio 5 : 9 represents	5	with antecedent = 5, consequent = 9.
	9

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

• Proportion:

The equality of two ratios is called proportion.If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.Here a and d are called extremes, while b and c are called mean terms.Product of means = Product of extremes.Thus, a : b :: c : d   (b x c) = (a x d).

• Fourth Proportional:

If a : b = c : d, then d is called the fourth proportional to a, b, c.Third Proportional:a : b = c : d, then c is called the third proportion to a and b.
Mean Proportional:Mean proportional between a and b is ab.

• Comparison of Ratios:

We say that (a : b) > (c : d)	a	>	c	.
	b		d

5. Compounded Ratio:

The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

6. Duplicate Ratios:

Duplicate ratio of (a : b) is (a² : b²).Sub-duplicate ratio of (a : b) is (a : b).Triplicate ratio of (a : b) is (a³ : b³).Sub-triplicate ratio of (a : b) is (a^1/3 : b^1/3).

If	a	=	c	, then	a + b	=	c + d	.     [componendo and dividendo]
	b		d		a - b		c - d

7. Variations:

We say that x is directly proportional to y, if x = ky for some constant k and we write,x   y.We say that x is inversely proportional to y, if xy = k for some constant k and

we write, x	1	.
	y

 

PERCENTAGE

1. Concept of Percentage:

By a certain percent, we mean that many hundredths.Thus, x percent means x hundredths, written as x%.

To express x% as a fraction: We have, x% =	x	.
	100

Thus, 20% =	20	=	1	.
	100		5

To express	a	as a percent: We have,	a	=		a	x 100	%.
	b		b			b

Thus,	1	=		1	x 100	%	= 25%.
	4			4

 

2. Percentage Increase/Decrease:

If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is:

	R	x 100	%
	(100 + R)

If the price of a commodity decreases by R%, then the increase in consumption so as not to decrease the expenditure is:

	R	x 100	%
	(100 - R)

3. Results on Population:

Let the population of a town be P now and suppose it increases at the rate of R% per annum, then:

1. Population after n years = P		1 +	R		n
			100

2. Population n years ago =	P
	1 + R n 100

4. Results on Depreciation:

Let the present value of a machine be P. Suppose it depreciates at the rate of R% per annum. Then:

1. Value of the machine after n years = P		1 -	R		n
			100

2. Value of the machine n years ago =	P
	1 - R n 100

3. If A is R% more than B, then B is less than A by		R	x 100	%.
		(100 + R)

4. If A is R% less than B, then B is more than A by		R	x 100	%.
		(100 - R)