Problem Solving

NeilWatson wrote:A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas for a total of $6.30, what number of apples and bananas did the customer purchase.

A)10
B)11
C)12
D)13
E)14

Here's an approach where we test the POSSIBLE SCENARIOS.

FACT #1: (total cost of apples) + (total cost of bananas) = 630 CENTS
FACT #2: total cost of bananas is DIVISIBLE by 50, since each banana costs 50 cents.

Now let's start testing POSSIBLE scenarios.

Customer buys 1 apple.
1 apple costs 70 cents, which means the remaining 560 cents was spent on bananas.
Since 560 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 2 apples.
2 apple costs 140 cents, which means the remaining 490 cents was spent on bananas.
Since 490 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 3 apples.
3 apple costs 210 cents, which means the remaining 520 cents was spent on bananas.
Since 520 is NOT divisible by 50, this scenario is IMPOSSIBLE

Customer buys 4 apples.
4 apple costs 280 cents, which means the remaining 350 cents was spent on bananas.
Since 350 IS divisible by 50, this scenario is POSSIBLE
350 cents buys 7 bananas.
So, the customer buys 4 apples and 7 bananas for a total of 11 pieces of fruit

Answer: B

Cheers,
Brent

A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas for a total of $6.30, what number of apples and bananas did the customer purchase.

A)10
B)11
C)12
D)13
E)14

I should mention that we can't really solve this question using regular algebra.
If we let A = total cost of apples (in cents),
and let B = total cost of bananas (in cents),
we get the equation 70A + 50B = 630

In high school we learned that, if we're given 1 equation with 2 variables, we cannot find the value of either variable. However, if we restrict the variables to POSITIVE INTEGERS, then there are times when we can find the value of a variable if we're given 1 equation with 2 variables.

Here's a similar question from the Official Guide: https://www.beatthegmat.com/og-13-132-t117594.html

Cheers,
Brent

NeilWatson wrote:A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas for a total of $6.30, what number of apples and bananas did the customer purchase.

A)10
B)11
C)12
D)13
E)14

I know this can be solved pretty easily with algebra but can anyone show me how they might use the answer choices to answer this problem? or is this not a good candidate for that

Solution:

I think the algebra approach is a pretty sound method for solving this problem.

We are given that apples were sold for $0.70 each and that bananas were sold for $0.50 each. We can set up variables for the quantity of apples sold and the quantity of bananas sold.

b = quantity of bananas sold

a = quantity of apples sold

With these variables, it follows that:

0.7a + 0.5b = 6.3

We can multiply this equation by 10 to get:

7a + 5b = 63

You may now notice that we do not have any other information to set up a second equation as we sometimes do for problems with two variables. So, we must use what we have. Keep in mind that variables a and b MUST be positive whole numbers, because you can't purchase 1.4 apples, for example. You also may notice that "7" and "63" have a factor of 7 in common. Thus, we can move "5b" to the other side of the equation and scrutinize the new equation carefully:

5b = 63 - 7a

5b = 7(9 - a)

b = [7(9 - a)]/5

Remember that a and b MUST be positive whole numbers here. Thus, 5 must evenly divide into 7(9 - a). Since we know that 5 DOES NOT divide evenly into 7, it MUST divide evenly into (9 - a). We can ask the question: What must a equal so that 5 divides into 9 - a? The only value a can be is 4. And we can check this:

(9 - a)/5 = ?

(9 - 4)/5 = ?

5/5 = 1

Since we know a = 4, we can use that to determine the value for b.

b = [7(9 - 4)]/5

b = [7(5)]/5

b = 35/5

b = 7

Thus a + b = 4 + 7 = 11

The answer B