## Data Sufficiency

# Solving GMAT Quant problems involving ranges with inequalities

On the GMAT, what appears to be little more than a simple rounding task might be anything but. Here's a breakdown of solving GMAT Quant problems involving ranges with inequalities. Note that you could apply these principles to GMAT problem solving questions, but these types of questions are more likely to show up in Data Sufficiency than Problem Solving, so we'll focus on that here. Take a look at this tricky Data Sufficiency question, for instance:

Is the exact value of x/y less than 0.5?

(1)When x and y are rounded to the nearest ten, x is 30, and y is 70.

(2)When x and y are rounded to the nearest unit, x is 32, and y is 65.

In order to answer the question stem, we need to determine the range (the minimum and maximum values) of x/y. To do this, we also need to know the range of each of the variables x and y individually. Since visualising each range in your head can be difficult, the key step when working with ranges is to translate each statement into an inequality for each variable.

Looking at the first statement alone, we’re told that when x is rounded to the nearest ten, x is 30. Possible values of x include 31, 27, and 34.99, since all 3 values round to 30. We can write a general inequality that describes all possible values for x:

Ineq. 1: 25 ≤ x < 35.

Following a similar logic for y, we get the inequality below:

Ineq. 2: 65 ≤ y < 75.

Notice that neither x nor y will ever equal their upper boundaries. This is because if x were 35, it would round to 40 rather than 30; likewise, if y were 75, it would round to 80 rather than 70.

Once the ranges of the individual variables have been translated into inequalities, we can now try to find the range of the value of x/y to answer the question: Is the exact value of x/y less than 0.5?

We can answer this question by finding both the smallest and largest possible value for x/y. If both the smallest and largest possible values are less than 0.5, then value of x/y will be less than 0.5. However, if the possible values span across 0.5, then we cannot answer the question stem with certainty, and so we would have insufficient data.

To obtain the smallest value of any fraction, we need the smallest possible numerator with the largest possible denominator. This occurs when x = 25 and y is just slightly less than 75, so that x/y > 25/75 = 1/3. In other words, x/y‘s lower boundary is slightly greater than 1/3, so sometimes it can be less than 0.5.

The answer to the question stem is sometimes “Yes.” For the largest value of the fraction, we need the largest possible numerator with the smallest possible denominator. This occurs when x is just slightly less than 35, and y = 65. In this case, x/y < 35/65 ≈ 0.53. So x/y could be any number slightly less than 0.53. Hence, the answer to the question stem is sometimes “No.” Overall, we cannot answer the question stem with certainty, so we have insufficient data. The second statement can be approached in the same manner. First, translate the statement into two inequalities:

Ineq. 3: 31.5 ≤ x < 32.5

Ineq. 4: 64.5 ≤ y < 65.5.

We again look for the smallest and largest values of x/y. For the smallest value, x/y < 31.5/65 < 0.5; x/y must be less than 0.5, giving a “Yes” answer to the question stem. The largest value of x/y, however, can be anything slightly less than 32.5/64.5 = 0.503.

For instance, x/y could be 0.501, giving a “No” answer to the question stem. We again have insufficient data. If we take both statements combined, we need to combine the inequalities, taking the intersection of ineq. 1 and 3, and the intersection of 2 and 4:

Ineq. 5: 31.5 ≤ x < 32.5

Ineq. 6: 65 ≤ y < 65.5

We find that 31.5/65 < x/y < 32.5/65 = 0.5. Since both the lower and upper bounds are less than 0.5, every value of x/y must be less than 0.5. The answer to the question stem is always “Yes”, so we have sufficient data; the answer choice is therefore C.

_{This was a sample of the in-depth instruction that GMAT Tutor offers about solving Data Sufficiency questions in the GMAT Quant section. For complete and interactive lessons and online tutor support, subscribe to one of GMAT Tutor's top-rated GMAT prep plans. Commitment-free trials are available for seven days.}