# Samples

Each panel below shows a GMAT math sample problem with a breakdown of the slow/hard way and the fast/easy way to solve it:

### Sample Problem #1: Problem Solving

Albert will receive \$14,600 over the period of one year in equal monthly payments.
How much will he have received after three months?

### The Hard/Slow Way

Find the monthly payment by dividing by 12: \$14,600 / 12 = \$1220
Then multiply by three to find the three-month total: \$1220 x 3 = \$3660

### The Easy/Fast Way

Three months is one quarter of a year. Divide the annual amount by 4: \$14,600 / 4 = \$3660

The arithmetic involved in this fast/easy solution is one short division (dividing by 3).

### SAMPLE PROBLEM #2: PROBLEM SOLVING

Which of the following values of p results in an integer value for the expression (77 +p) / p:

A) 4
B) 5
C) 6
D) 7
E) 8

### The Hard/Slow Way

Plug each of the five suggested values into the given expression:

A) (77+4)/4 = 81/4…..Not an integer
B) (77+5)/5 = 82/5…..Not an integer
C) (77+6)/6 = 83/6…..Not an integer
D) (77+7)/7 = 84/7…..Bingo
E) (77+8)/8 = 85/8…..Not an integer

### The Easy/Fast Way

Think first. Do a small amount of algebra. Transform the given expresson like this: (77+p)/p = 77/p + p/p = 77/p + 1.

The +1 doesn’t affect the integer status of the result. So the question becomes: which of the following values of p makes 77 / p an integer?

Expressed differently, which value of p goes into 77? The answer is immediately seen to be 7 because 77 is 7 times 11.

### SAMPLE PROBLEM #3: DATA SUFFICIENCY

Given the equation 30 + x = y – 42, which statement below would be sufficient to determine the value of y?

A) x = 3
B) y/x = 25
C) A and B are each sufficient on their own
D) A and B are sufficient only when combined
D) None of the above

### The Hard/Slow Way

A) Plugging in x=3 transforms the equation into 30+3 = y – 42 or y=42+30+3 =75. Conclusion: the information is sufficient.

B) Given y/x=25, we have y=25x. Substituting this, the original equation becomes 30+x = 25x – 42 or24x=72. This gives x=3, and so the equation can be solved for y in Statement B.

Conclusion: C.

### The Easy/Fast Way

My student solves this problem at a glance, immediately recognizing that each of the two statements provides sufficient information to answer the root question: what is the value of y? My student’s approach is explained in detail below.

A) The equation given in the root is a linear equation in x and y. When a value of x is given, the equation becomes a linear equation one unknown, which has a solution; so the information is sufficient.

B) The equation given in the root is a linear equation in x and y, and the equation y/x=25, or y=25x, is another linear equation in two unknowns. Two independent linear equations in two unknowns determine a unique solution. So the information is sufficient.