Ratio

RATIO AND PROPORTION

I. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b 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 or consequent.





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

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.



2. 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).



3. (i) Fourth Proportional: If a : b = c: d, then d is called the fourth proportional

to a, b, c.

(ii) Third Proportional: If a: b = b: c, then c is called the third proportional to

a and b.

(iii) Mean Proportional: Mean proportional between a and b is square root of ab



4. (i) COMPARISON OF RATIOS:

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

(ii) COMPOUNDED RATIO:

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



5. (i) Duplicate ratio of (a : b) is (a² : b²).

(ii) Sub-duplicate ratio of (a : b) is (√a : √b).

(iii)Triplicate ratio of (a : b) is (^a3 : ^b3).

(iv) Sub-triplicate ratio of (a : b) is (a ⅓ : b ⅓ ).

(v) If (a/b)=(c/d), then ((a+b)/(a-b))=((c+d)/(c-d)) (Componendo and dividendo)



6. VARIATION:

(i) We say that x is directly proportional to y, if x = ky for some constant k and

we write, x µ y.

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and

we write, x∞(1/y)

Examples:

Ex. 1. If a : b = 5 : 9 and b : c = 4: 7, find a : b : c.

Sol. a:b=5:9 and b:c=4:7= (4X9/4): (7x9/4) = 9:63/4

a:b:c = 5:9:63/4 =20:36:63.

Ex. 2. Find: (i) the fourth proportional to 4, 9, 12;

(ii) the third proportional to 16 and 36;

(iii) the mean proportional between 0.08 and 0.18.

Sol. i) Let the fourth proportional to 4, 9, 12 be x.

Then, 4 : 9 : : 12 : x ó4 x x=9x12 ó X=(9 x 12)/14=27;

Fourth proportional to 4, 9, 12 is 27.

(ii) Let the third proportional to 16 and 36 be x.

Then, 16 : 36 : : 36 : x ó16 x x = 36 x 36 ó x=(36 x 36)/16 =81

Third proportional to 16 and 36 is 81.

(iii) Mean proportional between 0.08 and 0.18

Ö0.08 x 0.18 =Ö8/100 x 18/100= Ö144/(100 x 100)=12/100=0.12

Ex. 3. If x : y = 3 : 4, find (4x + 5y) : (5x - 2y).

Sol. X/Y=3/4 ó (4x+5y)/(5x+2y)= (4( x/y)+5)/(5 (x/y)-2) =(4(3/4)+5)/(5(3/4)-2)

=(3+5)/(7/4)=32/7

Ex. 4. Divide Rs. 672 in the ratio 5 : 3.

Sol. Sum of ratio terms = (5 + 3) = 8.

First part = Rs. (672 x (5/8)) = Rs. 420; Second part = Rs. (672 x (3/8)) = Rs. 252.

Ex. 5. Divide Rs. 1162 among A, B, C in the ratio 35 : 28 : 20.

Sol. Sum of ratio terms = (35 + 28 + 20) = 83.

A's share = Rs. (1162 x (35/83))= Rs. 490; B's share = Rs. (1162 x(28/83))= Rs. 392;

C's share = Rs. (1162 x (20/83))= Rs. 280.

CHAIN RULE

_IMPORTANT FACTS AND FORMULAE

1. Direct Proportion: Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same

Ex. 1. Cost is directly proportional to the number of articles.

(More Articles, More Cost)



Ex. 2. Work done is directly proportional to the number of men working on it

(More Men, More Work)

2. Indirect Proportion: Two quantities are said to be indirectly proportional,if on the increase of the one, the other decreases to the same extent and vice-versa.

Ex. 1. The time taken by a car in covering a certain distance is inversely proportional to the speed of the car.

(More speed, Less is the time taken to cover a distance)



Ex. 2. Time taken to finish a work is inversely proportional to the num of persons working at it.

(More persons, Less is the time taken to finish a job)



Remark: In solving questions by chain rule, we compare every item with the term to be found out.



Examples:

Ex. 1. If 15 toys cost Rs, 234, what do 35 toys cost?

Sol. Let the required cost be Rs. x. Then,

More toys, More cost (Direct Proportion)

. 15 : 35 : : 234 : x <==> (15 * x) = (35 * 234) <==> x=(35 * 234)/15 = 546

Hence, the cost of 35 toys is Rs. 546.

Ex. 2. If 36 men can do a piece of work in 25 hours, in how many hours will 15 men

do it ?

Sol. Let the required number of hours be x. Then,

Less men, More hours (Indirect Proportion)

15 : 36 : : 25 : x <==> (15 * x) = (36 x 25) <==> (36 * 25)/15 = 60

Hence, 15 men can do it in 60 hours.

Ex. 3. If the wages of 6 men for 15 days be Rs.2100, then find the wages of

for 12 days.

Sol. Let the required wages be Rs. x.

More men, More wages (Direct Proportion)

Less days, Less wages (Direct Proportion)



Men 6 : 9 : : 2100 : x Days 15:12

Let the required consumption of coal be x units.

Less engines, Less coal consumed (direct proportion)

More working hours, More coal consumed (direct proportion)

Less rate of consumption, Less coal consumed(direct prportion)



Number of engines 9: 8

Working hours 8 : 13 :: 24 : x

Rate of consumption (1/3):(1/4)

[ 9 * 8 * (1/3) * x) = (8 * 13 * (1/4) * 24 ) <==> 24x = 624 <==> x = 26.

Hence, the required consumption of coal = 26 metric tonnes.