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?
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.
To solve this problem, we can use algebraic equations. Let's assign variables to each unknown value.
Let:
- the number of mangoes be M
- the number of honeydews be H
- the number of watermelons be W
Given information:
- 5/9 of the fruit bought were mangoes, so M/(M+H+W) = 5/9
- 1/3 of the fruit bought were honeydews, so H/(M+H+W) = 1/3
- the rest were watermelons, so W/(M+H+W) = 1 - (5/9 + 1/3) = 1/9
Now let's solve the equations step-by-step:
(a) Find the ratio of the number of mangoes to the ratio of the honeydews to the ratio of the watermelons:
We know that M/(M+H+W) = 5/9, H/(M+H+W) = 1/3, and W/(M+H+W) = 1/9.
So the ratio of the number of mangoes (M), honeydews (H), and watermelons (W) is:
M:H:W = (5/9):(1/3):(1/9) = 15:3:1
Therefore, the ratio of the number of mangoes to the ratio of the honeydews to the ratio of the watermelons is 15:3:1.
(b) How many fruits did he buy altogether?
To find the total number of fruit, we need to add up the quantities of mangoes, honeydews, and watermelons. So:
M + H + W = (15x) + (3x) + (x) = 19x
We know that a mango costs $1.50, which means each mango is worth 1.5 dollars. And we also know that mangoes cost $24 more than watermelons. Therefore:
(1.5)M = (1/2)(1.5)W + $24
Multiplying both sides by 2:
3M = 1.5W + 48
Since the ratio of M:W is 15:1 (from part a), we can simplify the equation:
3(15x) = 1.5(x) + 48
45x = 1.5x + 48
45x - 1.5x = 48
43.5x = 48
x = 48/43.5
x ≈ 1.1034
Now we can find the total number of fruit:
M + H + W = 19x
M + H + W = 19(1.1034)
M + H + W ≈ 20.9686
Since we can't have a fraction of a fruit, we round it to the nearest whole number.
M + H + W ≈ 21
Therefore, Joel bought approximately 21 fruits altogether.