Tricks to Solve Ratio and Proportion (Important Facts and Formula)

Tricks to Solve Ratio and Proportion (Important Facts and Formula) for Bank, SSC, Railway and Other Government Job related Competitive Exam. Ratio and Proportion questions are commonly asked in all the Competitive Exam. In this Article we will Discuss some Tricks to Solve Ratio and Proportion Questions Quickly.

1. 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 : b ).
(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)

7.A:B  & B:C Then A:B:C = ?

8.A:B  & B:C  & C:D Then A:B:C:D = ?

SOLVED PROBLEMS

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  ó 4 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.
Ex. 6. A bag contains 50 p, 25 P and 10 p coins in the ratio 5: 9: 4, amounting to
Rs. 206. Find the number of coins of each type.

Sol. Let the number of 50 p, 25 P and 10 p coins be 5x, 9x and 4x respectively.
(5x/2)+( 9x/ 4)+(4x/10)=206ó 50x + 45x + 8x = 4120ó1O3x = 4120 óx=40.

Number of 50 p coins = (5 x 40) = 200; Number of 25 p coins = (9 x 40) = 360;
Number of 10 p coins = (4 x 40) = 160.

Ex. 7. A mixture contains alcohol and water in the ratio 4 : 3. If 5 litres of water is added to the     mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture

Sol. Let the quantity of alcohol and water be 4x litres and 3x litres respectively
4x/(3x+5)=4/5 ó20x=4(3x+5)ó8x=20 óx=2.5
Quantity of alcohol = (4 x 2.5) litres = 10 litres.

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