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Operations with Integers

This topic has 4 expert replies and 0 member replies
danielanassar Newbie | Next Rank: 10 Posts Default Avatar
Joined
02 May 2016
Posted:
2 messages

Operations with Integers

Post Wed Jul 06, 2016 4:20 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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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    Post Wed Jul 06, 2016 5:13 pm
    Quote:
    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

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    Post Wed Jul 06, 2016 5:13 pm
    Quote:
    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: http://www.beatthegmat.com/og-13-132-t117594.html

    Cheers,
    Brent

    _________________
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    Come see all of our free resources

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    Post 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

    GMAT assassins aren't born, they're made,
    Rich

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    Post Thu Jul 07, 2016 2:57 pm
    I'm going to leave you guys in the shade here Very Happy

    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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