Till previous chapter of proportion, we have had only 2 variables in questions. What if we have more than 2 variables?
In this lesson, we will discuss the method to solve ratios with more than two variables.
Concept 1: If we have three variables.
If the ratio between first and second quantity is p:q and between second and third quantity is r:sthen what will be the ratio among first,second, and third quantity?
Mathematically if first, second, and third quantities are a,b, and c respectively then:
a:b = p:q
b:c = r:s
a:b:c =?
to solve this, we follow the procedure shown below: Following the arrows and multiplying the terms we get,
a:b:c =>pr: qr: qs

Concept 2: If we have four variables.
If the ratio between first and second quantity is p:q and between second and third quantity is r:s, between third and fourth is t:uthen what will be the ratio among first,second, third,and fourth quantity?
Mathematically if first, second, third,and fourth quantities are a,b, c, and d respectively then,
a:b = p:q
b:c = r:s
c:d =t:u
a:b:c:d =?
to solve this, we follow this procedure:
(Note: Just like the procedure shown in concept 1, we can extrapolate it to include the 3rd ratio, drawing alternate arrows in upward and downward directions.) Following the colors and multiplying the variables
a:b:c:d will be =>prt:qrt:qst:qsu
To test both the concepts studied above, solve the exercise given below:

EXERCISE:

Question 1: If A: B = 3: 4, B: C = 5: 7 and C: D = 8: 9 then A: D is equal to

(1) 3: 7

(2) 7: 3

(3) 21: 10

(4) 10: 21

Solution: (4)  Question 2: If a: b = , b: c = and d: c = then a:b: c: d is

(1)  4: 6: 7: 9

(2)  16: 24: 30: 35

(3)  8: 12: 15: 7

(4)  30: 35: 24: 16

Solution: (2)

Now  Question 3: If a: b = 5: 7 and c: d = 2a: 3b, then ac: bd is:

(1) 20: 38

(2) 50: 147

(3) 10: 21

(4) 50: 151

Solution: (2)

We have, Question 4: If A: B= , B: C = and C: D = , then the ratio A: B: C: D is?

(1)  6: 4: 8: 10

(2)  6: 8: 9: 10

(3)  8: 6: 10: 9

(4)  4: 6: 8: 10

Solution: (3)  Question 5: If A: B: C = 2: 3: 4, then ratio is equal to

(1) 8: 9: 16

(2) 8: 9: 12

(3) 8: 9: 24

(4) 4: 9: 16

Solution: (3)

A: B: C = 2: 3: 4 Question 6: If a, b, c are three numbers such that a: b = 3: 4 and b: c = 8: 9, then a: c is equal to?

(1) 1: 3

(2) 2: 3

(3) 3: 2

(4) 1: 2

Solution: (2)

We have a: b = 3: 4 = 6: 8

and b: c = 8: 9

Hence a: b: c = 6: 8: 9. So a: c = 6: 9 or 2: 3.

Question7:  If A: B = 2: 3 and B: C = 4: 5, then A: B: C is

(1) 2: 3: 5

(2) 5: 4: 6

(3) 6: 4: 5

(4) 8: 12: 15

Solution: (4)

A: B = 2: 3

B: C = 4: 5

A: B: C = 2 × 4: 3 × 4: 3 × 5

= 8: 12: 15

Question 8: If A: B = 2: 3, B: C = 2: 4 and C: D = 2: 5, then A: D is equal to:

(1) 2: 15

(2) 2: 5

(3) 1: 5

(4) 3: 5

Solution: (1)

We have A: B = 2: 3 = 4: 6

B: C = 2: 4 = 6: 12

C: D = 2: 5 = 12: 30

Hence A: B: C: D = 4: 6: 12: 30

Hence A: D = 4: 30 or 2: 15.

Question 9: The ratio of A to B is 4: 5 and that of B to C is 2: 3. If A equals 800, then C equals?

(1) 1000

(2) 1200

(3) 1500

(4) 2000

Solution: (3)

A: B = 4: 5

B: C = 2: 3

A: B: C = 4 × 2: 5 × 2: 5 × 3=8: 10: 15

If A equals 800, then C equals 1500.

Question 10: If A and B are in the ratio 3: 4, and B and C in the ratio 12: 13, then A and C will be in the ratio:

(1) 3:13

(2) 9: 13

(3) 36: 13

(4) 13: 9

Solution: (2)

We have A: B = 3:4 = 9: 12

B: C = 12: 13 A: B: C = 9: 12: 13

Hence A: C is 9: 13.

Question 11: If A: B = 3: 2 and B: C = 3: 4 then A: C is equal to?

(1) 1: 2

(2) 2: 1

(3) 8: 9

(4) 9: 8

Solution: (4)

We have A: B = 3: 2

B: C = 3: 4 A: B: C = 3 × 3: 2 × 3: 2 × 4

= 9: 6: 8 A: C = 9: 8

Question 12: If A: B is 2: 3, B: C is 6: 11, then A: B: C is:

(1) 2: 3: 11

(2) 4: 6: 22

(3) 4: 6: 11

(4) 2: 6: 11

Solution: (3)

A: B = 2: 3 = 4: 6

B: C = 6: 11 A: B: C = 4: 6: 11

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