The given question is a challenging GMAT 700+ level quant problem solving question combining concepts in linear equations and elementary number properties.

Question 19: Susan invited 13 of her friends for her birthday party and created return gift hampers comprising one each of $3, $4, and $5 gift certificates. One of her friends did not turn up and Susan decided to rework her gift hampers such that each of the 12 friends who turned up got $13 worth gift certificates. How many gift hampers did not contain $5 gift certificates in the new configuration?

- 5
- 2
- 9
- 3
- 7

INR

Total dollar value of all gift certificates in the initial configuration : 13 hampers × 12 dollars = $156

Total dollar value of all gift certificates in the new configuration = 12 hampers × 13 dollars = $156

No change in total value of gift certificates.

Susan started with 13 hampers.

So, she has 13 of $3, 13 of $4, and 13 of $5 certificates.

In the new configuration, $13 can be achieved as follows: {3, 3, 3, 4}, {4, 4, 5}, and {5, 5, 3}

The hampers that contain the configuration {3, 3, 3, 4} will not have $5 certificates.

Let Susan make x hampers of {3, 3, 3, 4}, y hampers of {4, 4, 5}, and z hampers of {5, 5, 3}.

The answer to ‘x’ is the answer to the question.

Number of $3 certificates will be 3x + z = 13 ....(1)

Number of $4 certificates will be x + 2y = 13 ....(2)

Number of $5 certificates will be y + 2z = 13 ....(3)

**Multiply (3) by 2**: 2y + 4z = 26

__Subtract (2) from (3) × 2__

2y + 4z = 26

-x - 2y = -13

------------------

4z – x = 13 ....(4)

------------------

**Multiply (1) by 4**: 12x + 4z = 52

__Subtract (4) and (1) × 4__

12x + 4z = 52

-4z + x = -13

----------------

13x = 39

-----------------

Or x = 3

'x' is the number of gift hampers without $5 certificates.

As x = 3, three gift hampers did not contain $5 gift certificates.

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