Help me in math

Question

A fruit vendor had a certain number of oranges and apples. Initially, for every 6 oranges, there were 5 apples. Later in the day, 24 oranges and 23 apples were sold. After these sales, it turned out that the number of apples left was exactly two-thirds the number of oranges left.

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Can you figure out how many apples remained at the end?

Answer 1

5/6 ratio of apples to oranges changes to 2/3

when 24 oranges are sold and 23 apples are sold

a/o = 5/6

cross multiply to get

6a = 5o

(a-23)/(o-24) = 2/3

cross multiply to get

3a -69 = 2o-48

3a = 2o +21

double that

6a =4o +42 subtract 6a = 5o eliminate one variable

0 = -o +42

o = 42 oranges originally

6a =5o = 5(42) = 210

a = 210/6 = 70/2 =35 apples originally

Answer 2

Here is one way to solve this problem. Let x represent the constant of proportionality between the oranges and apples. What that means is:

Number of oranges currently on hand is 6x.

Number of apples currently on hand is 5x.

For example if x = 2, the Oranges is 12 and Apples is 10 (6 to 5).

If x = 5, then O = 30 and A = 25 (still 6 to 5).

Now what needs to be done is to set up an equation in terms of x that models the situation.

After selling 24 oranges the number of oranges on hand is 6x - 24.

The number of apples on hand is 5x - 23.

At this point the number of apples IS 2/3 of the number of oranges:

5x - 23 = (2/3)(6x - 24)

To solve that equation, notice that the 3 in the denominator of the fraction 2/3 divides evenly into each term inside the parentheses. So go ahead and distribute the (2/3).

5x - 23 = 4x - 16

5x - 4x = 23 - 16

x = 7

Knowing the value of x means:

Number of oranges originally is 7(6) = 42.

Number of apples originally is 7(5) = 35.

Check (after sale of fruit):

Oranges: 42 - 24 = 18

Apples: 35 - 23 = 12

12 is 2/3 of 18.