For example:

20 meters to 15 meters to 25 meters

= 20 : 15 : 25

= 4 : 3 : 5

Write the following quantities in the form a : b : c.

(a) 8 liters, 16 liters, 6 liters

(b) 0.75 days : 30 hours : 9 hours

8 liters : 16 liters : 6 liters

= 8 : 16 : 6

= 4 : 8 : 3

0.75 days : 30 hours : 9 hours

= 0.75 × 24 hours : 30 hours : 9 hours

= 18 hours : 30 hours : 9 hours

= 6 : 10 : 3

A common error is to forget to convert days into hours before working out the solution.

(a) Multiplying all 3 quantities in the smaller ratio by the same number.

(b) Dividing all 3 quantities in the larger ratio by their highest common factor (HCF).

Determine whether the following are equivalent or not.

(a) 1 : 3 : 5, 3 : 9 : 15

(b) 2 : 3 : 6, 4 : 6 : 16

1 : 3 : 5

= 1 × 3 : 3 × 3 : 5 × 3

= 3 : 9 : 15

Therefore, 1 : 3 : 5 and 3 : 9 : 15 are both are equivalent.

2 : 3 : 6

= 2 × 2 : 3 × 2 : 6 × 2

= 4 : 6 : 12

Therefore, 2 : 3 : 6 and 4 : 6 : 16 are not are equivalent.

Determine whether the following are equivalent or not.

(a) 3 : 4 : 5, 9 : 12 : 15

(b) 2 : 8 : 10, 6 : 16 : 20

9 : 12 : 15

= 9 ÷ 3 : 12 ÷ 3 : 15 ÷ 3

= 3 : 4 : 5

Therefore, 3 : 4 : 5 and 9 : 12 : 15 are equivalent.

6 : 16 : 20

= 6 ÷ 2 : 16 ÷ 2 : 20 ÷ 2

= 3 : 8 : 10

Therefore, 2 : 8 : 10 and 6 : 16 : 20 are not are equivalent.

3 : 9 : 15

= 3 ÷ 3 : 9 ÷ 3 : 15 ÷ 3

= 1 : 3 : 5

If x : y : z = 1 : 5 : 8, then,

x : y = 1 : 5

y : z = 5 : 8

x : z = 1 : 8

John, Joe and Joseph have 12, 21 and 18 marbles respectively. Find the ratio of

(a) number of John's marbles to number of Joseph's marbles.

(b) number of Joe's marbles to number of Joseph's marbles.

(c) number of John's marbles to the difference in the number of marbles between Joe and Joseph.

(d) total number of marbles owned by John and Joe to number of Joseph's marbles.

The number of John's marbles to the number of Joseph's marbles

= 12 :18

= 2 : 3

The number of Joe's marbles to the number of Joseph's marbles

= 21 : 18

= 7 : 6

The number of John's marbles to the difference in the number of marbles between Joe and Joseph

= 12 : 21 – 18

= 12 : 3

= 4 : 1

The total number of marbles owned by John and Joe to the number of Joseph's marbles

= 12 + 21 : 18

= 33 : 18

= 11: 6

If x : y = 1 : 3 and y : z = 3 : 5, find x : y : z

Since the LCM of both 'y' is 1, just concatenate them together.

x : y : z

= 1 : 3 : 5

If x : y = 3 : 7 and y : z = 3 : 2, find x : y : z

Given that x : y = 3 : 7 and y : z = 3 : 2, the LCM of 7 and 3 is 21.

Hence,

x : y = 3 × 3 : 7 × 3 = 9 : 21

y : z = 3 × 7 : 2 × 7 = 21 : 14

Therefore, x : y : z = 9 : 21 : 14.

Given the following, x : y : z = 1 : 3 : 5 and y = 15, find the value of x and z.

x : y : z = 1 : 3 : 5

From the above, y = 3 parts.

Hence,

x = 1 part, so x = 15

z = 5 parts, so z = 5 × 5 = 25

(a) The ratio and the sum of the 3 quantities are given.

(b) The ratio and the difference between 2 of the 3 quantities are given.

Given that a : b : c = 6 : 5 : 7 and a + b + c = 198,

Given the following, p : q : r = 3 : 8 : 5 and the value of q – r = 36, find the values of p, q and r.

Given that x : y : z = a : b : c

x : (x + y + z) = a : (a + b + c)

y : (x + y + z) = b : (a + b + c)

z : (x + y + z) = c : (a + b + c)

Given the following, a : b : c = 3 : 6 : 4 and the value of b - c = 50, find the values of a + b + c.

Given that x : y : z = a : b : c

(x + y + z) : (x – y) = (a + b + c) : (a – b)

(x + y + z) : (x – z) = (a + b + c) : (a – c)

(x + y + z) : (y – z) = (a + b + c) : (b – c)

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