Flori’s Flower shop contains fewer than 150 flowers. All the flowers are purple, yellow, red or white.The ratio of purple flowers to yellow flowers is1 : 2, the ratio of yellow flowers to red flowers is3 : 4and the ratio of red flowers to white flowers is5 : 6.How many flowers are there in Flori’s shop?
Let's use algebra to solve this problem. Let P be the number of purple flowers, Y be the number of yellow flowers, R be the number of red flowers, and W be the number of white flowers.
From the given ratios, we know that:
- P : Y = 1 : 2
- Y : R = 3 : 4
- R : W = 5 : 6
We can use the first two ratios to write Y in terms of P, and then use the third ratio to write W in terms of R:
- P : Y = 1 : 2
- Y : R = 3 : 4
Multiplying both ratios gives:
- P : Y : R = 1 : 2 : 4
So we can write:
- P = x
- Y = 2x
- R = 4x
Now, using the third ratio, we can write:
- R : W = 5 : 6
Multiplying both sides by 4x (the value of R) gives:
- W = 4x * 6/5 = 24x/5
Finally, we know that the total number of flowers is:
- P + Y + R + W < 150
Substituting the values we found for P, Y, R, and W gives:
- x + 2x + 4x + 24x/5 < 150
- 31x/5 < 150
- x < 150*5/31
- x < 30
Since x is the smallest number of flowers (corresponding to purple flowers), we can conclude that there are less than 30 purple flowers, and therefore less than 60 yellow flowers, less than 120 red flowers, and less than 96 white flowers. The smallest possible total number of flowers is when each of these is just one less than the maximum, giving a total of 29 + 58 + 119 + 95 = 301 flowers. However, we need to check that this total satisfies the initial inequalities:
- P + Y + R + W < 150
- 29 + 58 + 119 + 95 < 150
This inequality is true, so our answer is:
Flori's flower shop contains 301 flowers.