RATIO AND PROPORTION

IMPORTANT FACTS AND FORMULAE

 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)

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): (7×9/4) = 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=9×12 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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