At a Youth Festival concert, the ratio of the number of Chinese to that of Malays was 8:7. The ratio of the number of Malays to that of Indians was 5:4.
a) Find the ratio of the number of Chinese to the number of Malays to the number of Indians.
b) After 39 Indians left the concert, 1/6 of the remaining audience at the concert were Indians. Find the total number of Chinese and Indians at the concert at first.
C : M = 8 : 7 = 40 : 35
M : I = 5:4 = 35 : 28
a)
C : M : I = 40 : 35 : 28
b) total at concert = 40x + 35x + 28x = 103x
number of Indians = 28x
after 39 left, number of Indians = 28x - 39
number left at concert = 103x - 39
28x - 39 = (1/6)(103x - 39)
168x - 234 = 103x - 39
65x = 195
x = 3
so
C = 120
M = 105
I = 84, for a total of 309
after departure: Indians --- 45
total after departure = 309-39 = 270
45/270 = 1/6, YEAHH!
C:M = 8:7 = 40:35
M:I = 5:4 = 35:28
C:M:I = 40:35:28
so, C:M = 40:35 = 8:7
I-39 = 1/6 (C+M+I-39)
I-39 = 1/6 (10/7 I + 5/4 I + I - 39)
I = 84
C = 10/7 * 84 = 120
so, 120+84 = 204
a) To find the ratio of the number of Chinese to the number of Malays to the number of Indians, we need to determine the common ratio between all three groups.
1. Start with the given ratios between Chinese, Malays, and Indians:
- Chinese : Malays = 8 : 7
- Malays : Indians = 5 : 4
2. To find the ratio of Chinese to Malays to Indians, we need to find a common value for Malays in both ratios. We can do this by multiplying the first ratio by 5 and the second ratio by 7:
Chinese : Malays = 8 : 7
multiplied by 5:
5(Chinese) : 5(Malays) = 5(8) : 5(7)
40 : 35
Malays : Indians = 5 : 4
multiplied by 7:
7(Malays) : 7(Indians) = 7(5) : 7(4)
35 : 28
3. Now we have the ratios:
Chinese : Malays : Indians = 40 : 35 : 28
Therefore, the ratio of the number of Chinese to the number of Malays to the number of Indians is 40 : 35 : 28.
b) To find the total number of Chinese and Indians at the concert initially, we need to set up an equation based on the given information.
1. Let's assume the initial number of Chinese is represented by 40x (using the ratio from part a).
Therefore, the initial number of Malays would be 35x and the initial number of Indians would be 28x.
2. After 39 Indians left, the number of remaining Indians is (28x - 39).
3. It is given that 1/6 of the remaining audience at the concert are Indians, so we can set up an equation:
(28x - 39) = (1/6)(40x + 28x)
4. Simplifying the equation:
(28x - 39) = (1/6)(68x)
6(28x - 39) = 68x
168x - 234 = 68x
168x - 68x = 234
100x = 234
x = 234/100
x = 2.34
5. Since x represents the initial number of Chinese (40x), we can determine the total number of Chinese and Indians initially:
Number of Chinese = 40x = 40(2.34) = 93.6 (rounded to the nearest whole number)
Number of Indians = 28x = 28(2.34) = 65.52 (rounded to the nearest whole number)
The total number of Chinese at the concert initially is approximately 94, and the total number of Indians at the concert initially is approximately 66.