# GMAT 700 800 Quant Question #19 | Algebra

###### GMAT Sample Questions | Linear Equations, Numbers Practice Question

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? 1. 5 2. 2 3. 9 4. 3 5. 7 ## Get to Q51 in GMAT Quant #### Online GMAT Course @ INR 3000 ### Video Explanation ## GMAT Live Online Classes #### Next Batch | October 4, 2020 ### Explanatory Answer | GMAT Algebra Practice Question #### KeyData 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.

#### Key Question: What are the possible combination of certificates in the new hampers?

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.

#### Assign Variables and Frame Linear Equations

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

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) #### Solve the 3 Equations 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.

##### Choice D is the correct answer.

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