Ratios of Three Quantities

Ratios of a to b to c that compares 3 quantities of the same unit are written as a : b : c.




For example:
20 meters to 15 meters to 25 meters

= 20 : 15 : 25

= 4 : 3 : 5


Note: Don't include the units in the simplified final answer. Writing the above as 4m : 3m: 5m is incorrect, because it's not actually magically changed to 4m, 3m and 5m. It's just how "large" or "small" they are relative to each other.


Example
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

Solution (a)
8 liters : 16 liters : 6 liters

= 8 : 16 : 6
= 4 : 8 : 3

Solution (b)
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.


Determining Given Ratios As Equivalent

The equivalence between 2 ratios of 3 quantities can be determined by:
(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).

Example 1 (Multiplying all 3 quantities by the same number)
Determine whether the following are equivalent or not.
(a) 1 : 3 : 5, 3 : 9 : 15
(b) 2 : 3 : 6, 4 : 6 : 16

Solution (a)
1 : 3 : 5
= 1 × 3 : 3 × 3 : 5 × 3
= 3 : 9 : 15

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

Solution (b)
2 : 3 : 6
= 2 × 2 : 3 × 2 : 6 × 2
= 4 : 6 : 12

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

Example 2 (Dividing all 3 quantities by HCF)
Determine whether the following are equivalent or not.
(a) 3 : 4 : 5, 9 : 12 : 15
(b) 2 : 8 : 10, 6 : 16 : 20

Solution (a)
9 : 12 : 15
= 9 ÷ 3 : 12 ÷ 3 : 15 ÷ 3
= 3 : 4 : 5

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

Solution (b)
6 : 16 : 20
= 6 ÷ 2 : 16 ÷ 2 : 20 ÷ 2
= 3 : 8 : 10

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


Simplifying A Ratio Of 3 Quantities To The Lowest Terms

The 3 quantities can be simplified by dividing or multiplying all 3 quantities by the same number. Division is typically when all 3 quantities are whole numbers, while multiplication is used when at least one of the quantities is a fraction.

Example 1 (Dividing all quantities)
3 : 9 : 15
= 3 ÷ 3 : 9 ÷ 3 : 15 ÷ 3
= 1 : 3 : 5

Example 2 (Multiplying all quantities)



Stating The Ratio Of Any 2 Quantities Given a Ratio Of 3 Quantities

If, for example, 3 quantities a : b: c are given, 2 quantities like a : b, b : c and a : c can be stated from it.

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

Example 2
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.

Solution (a)
The number of John's marbles to the number of Joseph's marbles
= 12 :18
= 2 : 3

Solution (b)
The number of Joe's marbles to the number of Joseph's marbles
= 21 : 18
= 7 : 6

Solution (c)
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

Solution (d)
The total number of marbles owned by John and Joe to the number of Joseph's marbles
= 12 + 21 : 18
= 33 : 18
= 11: 6


Given a :b and b : c, Find a : b : c

If a : b and b : c are given, then a : b : c can be found. The technique is to make sure that the value b in both a : b and b : c are of the same value. This can be achieved by determining the least common multiple, or LCM, and then multiplying all the values so that both values of b equal to the LCM.

Example 1 (Straightforward)
If x : y = 1 : 3 and y : z = 3 : 5, find x : y : z

Solution
Since the LCM of both 'y' is 1, just concatenate them together.
x : y : z
= 1 : 3 : 5

Example 2 (Using LCM)
If x : y = 3 : 7 and y : z = 3 : 2, find x : y : z

Solution
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.


Finding The Value Of Each Of The Three Quantities

When 3 quantities and the value of one of the quantities are given, the values of the other 2 quantities can be found using the unitary method.

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

Solution
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


Finding The Value Of Each Of The Three Quantities In A Ratio

Each of the 3 quantities can be found if:
(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.
Example 1 (Ratio and Sum of 3 Quantities Given)
Given that a : b : c = 6 : 5 : 7 and a + b + c = 198,

Solution


Example 2 (Ratio and Difference Between 2 of 3 Quantities Given)
Given the following, p : q : r = 3 : 8 : 5 and the value of q – r = 36, find the values of p, q and r.

Solution


Notes
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)


Finding The Sum Of Three Quantities Given The Ratio And The Difference Between Two Of The Three Quantities

If the ratio of three quantities and the difference between two of the three quantities are given, the sum of the three quantities can be found.

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

Solution


Notes
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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