A girl wishing to spend as much of her \(\$ 5\) as possible, i.e., reduce the amount of change, in a shop where she can buy burgers, fries or both. Can she decide what to buy? (1) She would get the same change from buying 4 burgers as she would from buying 6 portions of fries (2) The burgers are \(50 \%\) more expensive than the fries A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Linear equations form the backbone of many GMAT quantitative reasoning problems. Let's break this down a bit. A linear equation is essentially an equation that plots a straight line on a graph. It's written in the form \[ ax + by = c \], where \ a, b, \ and \ c\ are constants. For this type of problem, understanding how to set up and solve linear equations helps. Suppose you have two unknown variables like the prices of burgers \ (B)\ and fries \ (F) \. With the statements provided in the problem, we can create linear equations. First, when the girl buys 4 burgers or 6 fries, her change is the same. That gives us: \[ 5 - 4B = 5 - 6F \]
By simplifying, we reorganize as: \[ 4B = 6F \] This ability to form and manipulate linear equations is a crucial skill in solving GMAT questions efficiently.

Variable substitution is another powerful GMAT problem-solving strategy. This means replacing one variable with an expression involving another variable. For instance, from the given problem, we had two statements: \[ 4B = 6F \] and \[ B = 1.5F \]. Substituting the second equation into the first helps in verifying the consistency between variables. We substitute \[ B \] with \[ 1.5F \], resulting in: \[ 4(1.5F) = 6F \]. This simplifies to: \[ 6F = 6F \].
This demonstrates that our initial assumptions are consistent. Variable substitution helps in breaking down complex equations into simpler parts, thus making it easier to solve. When approaching GMAT problems, practice variable substitution whenever you have interconnected variables and see if that helps simplify the problem.

The GMAT quantitative section often tests your problem-solving skills through questions like these. It's essential to understand that it's not just about finding the answer but also about the logical steps leading there. Breaking down the problem: We are asked if the girl can decide what to buy with her $5. We have two conditions that help form equations. Statement 1 lets us form an equation about her change. Statement 2 tells us the relative price difference between a burger and fries. Analyzing individual statements can sometimes lead to a solution, but often, like in this problem, both statements together help verify the consistency rather than sufficiency. Practicing GMAT quantitative reasoning with various types of problems will help you become more adept at dissecting and solving them efficiently.

Mathematical consistency ensures that the relationships we deduce from equations hold true across different scenarios. In the context of the problem, once we derived the equations \[ 2B = 3F \] and \[ B = 1.5F \], we substituted one into the other to check consistency: \[ 2(1.5F) = 3F \]. The resulting \[ 3F = 3F \] confirms consistency because it is an identity. However, being consistent doesn't always mean solving the problem entirely. Here, while consistent, these statements don't give absolute values or directly answer the question of exact purchases to minimize change. So when solving GMAT problems, always check for both consistency and sufficiency. Both are critical in confirming if the derived relationships are logical and complete enough to answer the given problem fully.