Question

Joel bought some mangoes, honeydews and watermelons. 5/9 of the fruit bought were mangoes, 1/3 of the fruit bought wore honoydews and the rest
were watermelons. He paid $24 more for mangoes than watermelons. A mango costs 1/2 as much as a honeydew and a honeydew coats 1/2 as
much as a watermelon. A mango cost $1.50

(a) Find the ratio of the number of mangoes to the ratio of he honeydews to the ratio of the watermelons

(b) How many fruit did he buy altogether?

Answer 1

Let's start by finding the ratio of each type of fruit bought.

Let M = number of mangoes bought
H = number of honeydews bought
W = number of watermelons bought

From the problem, we know that:
M + H + W = total number of fruit bought

And we also know the following ratios:
M/(M + H + W) = 5/9
H/(M + H + W) = 1/3
W/(M + H + W) = 1 - (5/9) - (1/3) = 1/9

Simplifying these ratios, we get:
M/(M + H + W) = 5/9
H/(M + H + W) = 3/9
W/(M + H + W) = 1/9

We can also use the information given about the prices of the fruit to set up an equation involving the cost of the mangoes and the cost of the watermelons:
1.5M = (1/2)(1/2)(W) + 24

Simplifying this equation, we get:
1.5M = 0.25W + 24

Now we can solve for one variable in terms of the others. Let's solve for M in terms of W:
1.5M = 0.25W + 24
M = (0.25/1.5)W + (24/1.5)
M = (1/6)W + 16

Now we can substitute this expression for M into the ratio equation we found earlier and simplify:
M/(M + H + W) = 5/9
[(1/6)W + 16]/[(1/6)W + 16 + H + W] = 5/9
[(1/6)W + 16]/[(7/6)W + H + 16] = 5/9
45(1/6)W + 720 = 35(1/6)W + 5H + 80
10(1/6)W - 5H = -640

We also know that W/(M + H + W) = 1/9, so we can set up another equation:
W/(M + H + W) = 1/9
W/(W + (1/6)W + 16 + H) = 1/9
9W = 7.5W + 135 + 9H
1.5W - 9H = -135

Now we have two equations and two unknowns (W and H). We can solve this system of equations using substitution or elimination:

1.5W - 9H = -135 (multiply by 2 to eliminate decimal)
3W - 18H = -270

10(1/6)W - 5H = -640

Adding these two equations, we get:
13W - 23H = -910

Solving for W in terms of H by rearranging and substituting:
10(1/6)W - 5H = -640
10(1/6)W = 5H - 640
W = (5/10.167)H - (640/10.167)
W = 0.491H - 62.8

Substituting this expression for W into the equation we found earlier for the ratio:
[(1/6)W + 16]/[(7/6)W + H + 16] = 5/9
[(1/6)(0.491H - 62.8) + 16]/[(7/6)(0.491H - 62.8) + H + 16] = 5/9
0.221H + 8.7 = 0.442H + 28.4
0.221H = 19.7
H = 89

Now we can use this value of H to find the value of W using the expression we found earlier:
W = 0.491H - 62.8
W = 25.88

And we can find the value of M using the equation we found earlier:
M = (1/6)W + 16
M = 20.48

Finally, we can add up the total number of fruit bought:
M + H + W = 20.48 + 89 + 25.88 = 135.36, or approximately 135 (rounded to the nearest whole number)

(a) The ratio of the number of mangoes to the ratio of he honeydews to the ratio of the watermelons is:
M:H:W = 20.48:89:25.88 (rounded to two decimal places)

(b) Joel bought 135 fruit altogether.