First, the problem says there are a total of 187 fruits. These are apples, mangoes, and oranges. The ratio of apples to mangoes is 5:2. Hmm, okay, so for every 5 apples, there are 2 mangoes. We represent the number of apples and mangoes in terms of a variable using this ratio.
Let the number of apples be $2 and mangoes as 2x. That way, the ratio is maintained. So, apples = 5x, mangoes = 2x. Now, the total number of fruits is 187, so if I add apples, mangoes, and oranges together, it should equal 187. Let me write that down:
5x (apples) + 2x (mangoes) + oranges = 187
So, 7x + oranges = 187. That means oranges = 187 - 7x. Okay, so oranges are 187 minus 7 times x.
Now, the next part of the problem says that after selling some fruits, the ratio of unsold apples to unsold oranges becomes 3:2. She sells 75 apples, 26 mangoes, and half of the oranges. So, the number of unsold apples would be the original number minus 75, unsold mangoes would be original minus 26, and unsold oranges would be half of the original.
she sells half of the oranges, so the unsold oranges would be the other half. So, if she had O oranges, she sold O/2, so unsold is also O/2. But O is 187 - 7x, so unsold oranges would be (187 - 7x)/2.
Similarly, unsold apples would be 5x - 75, and unsold mangoes would be 2x - 26. But the problem doesn't mention the ratio involving mangoes, only apples and oranges. So, the ratio of unsold apples to unsold oranges is 3:2.
So, Setting up the equation: (5x - 75) / [(187 - 7x)/2] = 3/2
Hmm, let me write that more clearly:
(5x - 75) / [(187 - 7x)/2] = 3/2
To simplify this, I can multiply both sides by [(187 - 7x)/2] to get rid of the denominator:
5x - 75 = (3/2) * [(187 - 7x)/2]
Wait, actually, let me cross-multiply to make it easier. So, cross-multiplying gives:
2(5x - 75) = 3(187 - 7x)/2
Wait, no, cross-multiplying the ratio:
(5x - 75) / [(187 - 7x)/2] = 3/2
Which is the same as:
(5x - 75) 2 = 3 (187 - 7x)
Yes, that's better. So, expanding both sides:
2(5x - 75) = 3(187 - 7x)
Compute each side:
Left side: 10x - 150
Right side: 561 - 21x
So, bringing all terms to one side:
10x - 150 = 561 - 21x
Adding 21x to both sides:
31x - 150 = 561
Adding 150 to both sides:
31x = 711
So, x = 711 / 31
Compute that. 3123 is 713, which is too much. So, 3122 = 682, 711 - 682 = 29. So, 22 and 29/31. Hmm, that seems a bit messy. Wait, maybe I made a mistake in the cross-multiplication step. Let me check again.
Original ratio:
(5x - 75) / [(187 - 7x)/2] = 3/2
Cross-multiplying:
(5x - 75) 2 = 3 (187 - 7x)
Yes, that's correct. So, 10x - 150 = 561 - 21x
Adding 21x: 31x - 150 = 561
Adding 150: 31x = 711
So, x = 711 / 31. Let me divide 711 by 31.
31*20 = 620, 711 - 620 = 91
31*2 = 62, 91 - 62 = 29
So, 20 + 2 = 22, with a remainder of 29. So, x = 22 + 29/31. Hmm, that's a fractional value, which might be okay, but Wait, x should be an integer because the number of fruits can't be a fraction. So, maybe I made a mistake somewhere.
Let me go back through the steps.
Total fruits: 187
Apples:5x, Mangoes:2x, Oranges:187 -7x
After selling:
Apples sold:75, so unsold apples:5x -75
Mangoes sold:26, so unsold mangoes:2x -26
Oranges sold: half, so unsold oranges: (187 -7x)/2
Given that unsold apples / unsold oranges = 3/2
So, (5x -75) / [(187 -7x)/2] = 3/2
Which simplifies to:
(5x -75)2 = 3(187 -7x)
10x -150 = 561 -21x
10x +21x = 561 +150
31x = 711
x=711/31=22.935
