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Operations with Integers

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Operations with Integers

by danielanassar » Wed Jul 06, 2016 4:20 pm
Could you please help me understand how to solve this problem:

A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. if a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
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by Brent@GMATPrepNow » Wed Jul 06, 2016 5:13 pm
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

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by Brent@GMATPrepNow » Wed Jul 06, 2016 5:13 pm
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

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Brent
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by [email protected] » Wed Jul 06, 2016 9:06 pm
Hi danielanassar,

In the future, you should make sure to post the entire prompt (including the 5 answer choices). In many cases, the answer choices provide a 'hint' as to how you might go about solving the problem. Here, the answer choices are relatively small and 'close together', so we can use a bit of 'brute force' to get to the correct answer:

We know that there will be no fewer than 10 total pieces of fruit and no more than 14 total pieces of fruit that will total $6.30, so I'm going to list out the first several multiples of apple prices and banana prices:

Apples:
$0.70
$1.40
$2.10
$2.80
$3.50
$4.20
$4.90
$5.60
Etc.

Bananas:
$0.50
$1.00
$1.50
$2.00
$2.50
$3.00
$4.00
$4.50
Etc.

Now we just have to find a pair of numbers (one from each group) that will total $6.30. It's not too much work to find that $2.80 and $3.50 total $6.30, so the total number of pieces of fruit is 7+4 = 11

Final Answer: B

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by Matt@VeritasPrep » Thu Jul 07, 2016 2:57 pm
I'm going to leave you guys in the shade here :D

We know a + b < 13, since 13 * 50¢ is too much money.

We know a + b > 9, since 9 * 70¢ = $6.30, and we were told that we didn't only buy apples.

So a + b = 10, 11, or 12.

We also know that 7a + 5b = 63. If a and b are both odd, then we'd have Odd + Odd, which = Even. But we're told that 7a + 5b = Odd. Hence a = even and b = odd, or vice versa.

Since Even + Odd = Odd, we know the only answer is 11.
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by Jeff@TargetTestPrep » Tue Apr 10, 2018 10:34 am
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
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 number of apples sold and the number of bananas sold.

b = number of bananas sold

a = number 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

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 whole numbers, because you can't purchase 1.4 apples, for example. Notice also that 7 and 63 have a factor of 7 in common. Thus, we can move 7a and 63 to one side of the equation and leave 5b on 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? Of course, a could equal 9; but that would produce a zero for b and since the question states apples AND bananas were purchased, b cannot equal zero. The only other value a can be is 4. 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 of b.

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

b = [7(5)]/5

b = 35/5

b = 7

Thus a + b = 4 + 7 = 11.

Answer: B

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by GMATGuruNY » Wed Apr 11, 2018 2:02 am
A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14
Let $B = the amount spent on bananas and $A = the amount spent on apples.

Since each banana sells for 50 cents, we get the following options for $B, in cents:
50, 100, 150, 200...

Every value in the list above ends in 50 or 00.
Implication:
Since $B + $A = 630, $A must end in either 80 or 30.

Since each apple sells for 70 cents, we get the following options for $A:
70, 140, 210, 280...

Test the value in blue.
If $A = 280, then $B = 630-280 = 350.
In this case:
Number of apples purchased = (total spent on apples)/(price per apple) = 280/70 = 4.
Number of bananas purchased = (total spent on bananas)/(price per banana) = 350/50 = 7.
Total amount of fruit purchased = 4+7 = 11.

The correct answer is B.
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