How to Quickly Interpret Ranges of Variables | Algebra

Sometimes a GMAT Quant question will give us multiple ranges of values that apply to a single variable, and when this happens it can really take us for a ride. Evaluating these ranges to arrive at deductions can extremely confusing, so today we will look at some strategies for how to deal with such problems.

To start off, let’s take a look at an example problem:

If it is true that ${z}$ < 8 and 2 ${z}$ > -4, which of the following must be true?

(A) -8 < ${z}$ < 4
(B) ${z}$ > 2
(C) ${z}$ > -8
(D) ${z}$ < 4
(E) None of the above

Given that $z$ < 8 and 2 $z$ > -4, we know that $z$ > -2. This means -2 < $z$ < 8. $z$ must lie within that range, hence $z$ can take values such as -1, 0, 5, 7.4, etc.

Now, which of the given answer choices would hold true for ALL such values? Let’s examine each option and see:

(A) -8 < $z$ < 4
We know that $z$ may be more than 4, so this range does not hold true for all possible values of $z$ .

(B) $z$ > 2
We know that $z$ may be less than 2, so this also does not hold true for all possible values of $z$ .

(C) $z$ > -8
No matter what value $z$ will take, it will always be more than -8. This range holds true for all values of $z$ .

(D) $z$ < 4
We know that $z$ may be greater than 4, so this does not hold for all possible values of $z$ .

Our answer is (C).

To understand this concept more clearly, let’s use a real life example:

We know that Anna’s weight is more than 120 pounds but less than 130 pounds. Which of the following is definitely true about her weight?

(A) Her weight is 125 pounds.

(B) Her weight is more than 124 pounds.

(D) Her weight is more than 110 pounds.

Can we say that her weight is 125 pounds? No – we just know that it is more than 120 but less than 130. It could be anything in this range, such as 122, 125, 127.5, etc.

Can we say that her weight is more than 124 pounds? This may be true, but it might not be true. Knowing our given range, her weight could very well be 121 pounds, instead.

Can we say her weight is less than 127 pounds? Again, this might not necessarily be true. Her weight could be 128 pounds.

Now, can we say that her weight is more than 110 pounds? Yes – since we know Anna’s weight is between 120 and 130 pounds, it must be more than 110 pounds.

This question uses the same concept as the first question! If you look at that question again, it will hopefully make much more sense. Now try solving this example problem:

If ${\frac{1}{55}\ <\ x\ <\ \frac{1}{22}}$ and ${\frac{1}{33}\ <\ x\ <\ \frac{1}{11}}$ , then which of the following could be the value of ${x}$ ?

(i) ${\frac{1}{54}}$
(ii) ${\frac{1}{23}}$
(iii) ${\frac{1}{12}}$

(A) Only (i)
(B) Only (ii)
(C) (i) and (ii)
(D) (ii) and (iii)
(E) (i), (ii) and (iii)

In this problem, we are given two ranges of $x$ . We know that $\frac{1}{55}\ <\ x\ <\ \frac{1}{22}$ and $\frac{1}{33}\ <\ x\ <\ \frac{1}{11}$ , so $x$ is greater than $\frac{1}{55}$ AND it is greater than $\frac{1}{33}$ . Since $\frac{1}{33}$ is greater than $\frac{1}{55}$ (the smaller the denominator, the larger the number), we just need to know that $x$ will be greater than $\frac{1}{33}$ .

We are also given that $x$ is less than $\frac{1}{22}$ AND it is less than $\frac{1}{11}$ . Since $\frac{1}{22}$ is less than $\frac{1}{11}$ , we really just need to know that $x$ is less than $\frac{1}{22}$ .

Hence, the range for $x$ should be $\frac{1}{33}\ <\ x\ <\ \frac{1}{22}$ . $x$ could take all values that lie within this range, such as $\frac{1}{32}$ , $\frac{1}{31}$ , $\frac{1}{24}$ , $\frac{1}{23}$ , etc.

Looking at the answer choices, we can see that $\frac{1}{54}$ and $\frac{1}{12}$ (i and iii) are both out of this range. Therefore, our answer is (B).

If we go back to our real life example, this is what the question would look like now:

We know that Anna’s weight is more than 110 pounds but less than 130 pounds. We also know that her weight is more than 115 pounds but less than 140 pounds. Which of the following is definitely true about her weight?

(A) Her weight is 112 pounds.
(B) Her weight is 124 pounds.
(C) Her weight is 135 pounds.

We are given that Anna’s weight is more than 110 pounds and also more than 115 pounds. Since 115 is more than 110, we just need to know that her weight is more than 115 pounds. We are also given that Anna’s weight is less than 130 pounds and also less than 140 pounds. Since 130 is less than 140, we just need to know that her weight is less than 130 pounds.

Now we have the following range: 115 pounds < Anna’s weight < 130 pounds. Only answer choice B lies within this range, so that is our answer.

We hope you see that evaluating ranges of numbers on GMAT questions is not difficult when we consider them in terms of a real life example. The same logic that we use in the simple weight problem is also applicable when algebraic data is given.

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