Real Numbers - GMAT Quantitative

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Question

is a real number. True or false: is a rational number.

Statement 1: $N + \frac{6}{11}$ is an irrational number.

Statement 2: $\frac{5}{7}N$ is an irrational number.

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Answer

Assume Statement 1 alone. $\frac{6}{11}$ , the quotient of integers, is rational. The sum of any two rational numbers is rational, so if is rational, then $N + \frac{6}{11}$ is a rational number. However, $N + \frac{6}{11}$ is irrational, so, contrapositively, is irrational.

Assume Statement 2 alone. $\frac{5}{7}$ , the quotient of integers, is rational. The product of any two rational numbers is rational, so if is rational, then $\frac{5}{7}N$ is a rational number. However, $\frac{5}{7}N$ is irrational, so, contrapositively, is irrational.

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Question

$\small a + 8 = b+12$

$\small c+ 5 = d + 11$

$\small a + 4 = d + 3$

Order $\small a,b, c,d$ from least to greatest.

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Answer

We can find each in terms of .

$\small a + 8 = b+12$

$\small a + 4 = d + 3$

$\small \small c+ 5 = d + 11$

In ascending order, the four values are:

The correct choice is $\small b,a,d,c$ .

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Question

Is z a negative number?

(1) $\dpi{100} \small 13z>14z$

(2) $\dpi{100} \small 4+z$ is positive

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Answer

(1) Subtracting $\dpi{100} \small 13z$ from both sides of $\dpi{100} \small 13z>14z$ shows that $\dpi{100} \small 0>z$ , so $\dpi{100} \small z$ must be negative $\dpi{100} \small \rightarrow$ SUFFICIENT

(2) Subtracting 4 from both sides of $\dpi{100} \small 4+z>0$ gives $\dpi{100} \small z>-4$ . Therefore, for this statement, $\dpi{100} \small z$ can be negative but is not necessarily negative. $\dpi{100} \small \rightarrow$ NOT SUFFICIENT.

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Question

If , , and are points on the number line, what is the distance between and ?

(1) The distance between and is 7.

(2) The distance between and is 12.

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Answer

Looking at statements (1) and (2) separately, there is no information about the distance between and . Looking at statements (1) and (2) together, there are two possibilities: 1) and are on different sides of , and the distance between them is 19; 2) and are on the same side of , and the distance between them is 5. Therefore, we cannot get the only possible answer to this question based on the two statements.

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Question

Data sufficiency question- do not actually solve the question

How many children are attending a party?

1. If 6 more children attend the party, there will be 24 children there.

2. If 10 children leave the party, there will be less than 12 children in attendance.

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Answer

Statement 1 precisely tells how many children are attending the party while Statement 2 only provides a range.

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Question

A sandwich shop offers a discount in which customers pay $2.99 for each additional sandwich after purchasing their first sandwich. Emily bought her first sandwich for $4.95. If she purchased 5 additional sandwiches to take home, then the total amount she paid is equivalent to which of the following?

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Answer

The price of 6 sandwiches can be expressed as . Regrouping is needed since this is not in the answer choices. If $3.00 is used instead of $2.99, each of the five additional sandwiches is .01 too high and the total is .05 too high. Therefore, this amount needs to be subtracted from the $4.95.

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Question

Given that and , is positive, negative, or zero?

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Answer

If it is known that , it can be determined that $x = \frac{2}{3}y$ .

Then $x - y = \frac{2}{3} y - y = -\frac{1}{3} y$

Since , $x - y = -\frac{1}{3} y < 0$ , and the question is answered.

But if we only know that , it is not sufficient.

Case 1:

yields

Case 2:

yields

The answer is that Statement 1 is sufficient, but not Statement 2.

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Question

Which of the following is the decimal representation of an irrational number? (Assume any observed patterns continue)

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Answer

A number is rational if and only if its decimal representation is terminating or repeating.

and are terminating decimals, which are rational (note: the latter number is not equal to $\pi$ ). Both can be eliminated.

has a repeating digit and has a repeating group of five digits, and are therefore repeating decimals, which are also rational. Both can be eliminated.

has an infinite pattern - the number of zeroes that precede each one is increasing by one with each group - but since the pattern is not repeating, it is neither a terminating decimal nor a repeating decimal. It is irrational.

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Question

Jenkins wanted to find all the real solutions of the following expression. Find them for him.

I) has a vertex at the point .

II) is never undefined.

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Answer

To find the real solutions, we need to know what c is. An easy way to do that would be if we had a point on the function.

I) gives us such a point. Therefore, we are able to substitute in our values and solve for c using algebraic operations.

$7=144-36+c \rightarrow 7=108+c$

Thus our equation becomes:

.

From here we can try to factor to find the real solutions or you can use the quadratic formula.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Plugging in our values we get the solution .

II) Is not really relevant, we are dealing with a parabola, so it shouldn't be undefined anyway.

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Question

Evaluate $A^{B}$ .

Statement 1:

Statement 2:

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Answer

From Statement 1 alone, since 1 raised to the power of any real number is equal to 1, we know that

$A^{B} = 1 ^{B} = 1$ .

However, we cannot determine the value of $A^{B}$ knowing only that . If is a nonzero number, then $A^{B} = A^{0} = 1$ . However, if , then

$A^{B} = 0^{0}$ ,

which is an undefined expression.

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Question

True or false: is an integer.

Statement 1: The multiplicative inverse of is not an integer.

Statement 2: The additive inverse of is an integer.

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Answer

Assume Statement 1 alone. The multiplicative inverse of a number is the number which, when multiplied by that number, yields a product of 1. If the multiplicative inverse of is not an integer, it is possible for to be an integer or to not be one, as is shown in these examples:

If , which is an integer, then, since $2 \cdot \frac{1}{2} = 1$ , the multiplicative inverse of is $\frac{1}{2}$ , which is not an integer, If $N = \frac{2}{3}$ , which is not an integer, then, since $\frac{2}{3} \cdot \frac{3} {2} = 1$ , the multiplicative inverse of is $\frac{3} {2}$ , which is not an integer.

Assume Statement 2 alone. The additive inverse of a number is the number which, when added to that number, yields a sum of 0. If is the additive inverse of the number , then

, or

$N = -M = -1 \cdot M$

by Statement 2, is an integer; ., the product of integers, is itself an integer.

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Question

Evaluate $A \cdot (B + C)$ .

Statement 1:

Statement 2:

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Answer

Assume Statement 1 alone.

$A \cdot (B + C) = 0 \cdot (B + C) = 0$ , since 0 multiplied by any number yields a product of 0.

Assume Statement 2 alone.

$A \cdot (B + C) = A \cdot (0 + C) = A \cdot C$ , since 0 added to any number yields the sum 0. However, without knowing and , or their product, it is impossible to determine the result.

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Question

is a real number. True or false: is a rational number.

Statement 1: A square whose side has length has area $\frac{1}{4}$ .

Statement 2: $N \div 6$ is a rational number.

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Answer

From Statement 1 alone, since a side of a square with area $\frac{1}{4}$ has as its sidelength

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$ ,

$N= \frac{1}{2} = 1 \div 2$ .

This is the quotient of two integers; by definition, this is rational.

From Statement 2 alone, if we let $M = N \div 6$ , then $N = 6 \cdot M$ . is rational, and so is 6, so their product, , is also rational.

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Question

is a real number. True or false: is a rational number.

Statement 1: $N^{2} - 1$ is a rational number.

Statement 2: is a rational number.

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Answer

Statement 1 provides insufficient information to determine whether is rational or not.

If , which, being an integer, is rational, then $N^{2} - 1 = 2^{2} - 1 = 4 - 1 = 3$ , which, being an integer, is rational.

If $N = \sqrt{2}$ . which is irrational, then $N^{2} - 1 =\left ( \sqrt{2} \right )^{2} - 1 = 2 - 1 = 1$ , which, being an integer, is rational.

Statement 2 is, however, sufficient. If for some rational number , then . 1, being an integer, is rational, and as the difference of rational numbers, is rational.

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Question

is a real number. True or false: is a rational number.

Statement 1: $N^{3}-2N^{2}-2N + 4 = 0$

Statement 2: $N^{4} - 4 = 0$

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Answer

Assume Statement 1 alone. The polynomial expression in

$N^{3}-2N^{2}-2N + 4 = 0$

can be factored as follows:

$N^{2} (N-2)-2 (N-2) = 0$

$(N^{2}-2 ) (N-2) = 0$

Either , in which case , which, as an integer is also rational, or

$N ^{2}- 2 = 0$ , in which case:

$N^{2} = 2$

or either $N = - \sqrt{2}$ or $N = \sqrt{2}$ , both irrational.

Therefore, Statement 1 provides insufficient information.

Assume Statement 2 alone. If $N^{4} - 4 = 0$ , then $N^{4} = 4$ , making a fourth root of 4. A rational root of an integer must itself be an integer, and there is no integer which, when taken to the fourth power, is equal to 4. Therefore, must be irrational.

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Question

is a real number, with $N^{2}$ a positive integer. True or false: is a rational number.

Statement 1: $N^{2}$ is a multiple of 9.

Statement 2: $N^{2}$ is a multiple of 12.

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Answer

The two statements together provide insufficient information to answer the question. For example, 36 and 72 are both multiples of both 9 and 12. However, the positive square root of 36 is 6, making an integer and, consequently, rational. However, the positive square root of 72 is not an integer, since 72 falls between consecutive perfect squares 64 and 81 (the squares of 8 and 9, respectively), and any square root of an integer that is not itself an integer must be irrational.

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Question

is a real number. True or false: is an integer.

Statement 1: $N^{3} - 4N^{2}+ 4N -16 = 0$

Statement 2: $N^{3} - 4N^{2}- 4N +16 = 0$

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Answer

Assume Statement 1 alone. The polynomial expression in

$N^{3} - 4N^{2}+ 4N -16 = 0$

can be factored as follows:

$N^{2} (N - 4)+ 4 (N - 4) = 0$

$\LARGE (N^{2} + 4) (N - 4) = 0$

Either $\normal { N - 4 = 0 }$ , in which case , or

$N^{2}+ 4 = 0$ - which is impossible for any real value of , since

$N^{2} \ge 0$ , and

$N^{2}+ 4 \ge 4 > 0$

Therefore, is an integer.

Assume Statement 2 alone. Again, we factor:

$N^{3} - 4N^{2}- 4N +16 = 0$

$N^{2} (N - 4)- 4 (N - 4) = 0$

$(N^{2} - 4): (N - 4) = 0$

Similarly to the polynomial in Statement 1, , , or . All three possible values of are integers.

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Question

Evaluate $A \cdot B + C \cdot D$ .

Statement 1: $C \cdot D + A \cdot B = 66$

Statement 2: $B \cdot A + D \cdot C= 66$

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Answer

Assume Statement 1 alone. By the commutative property of addition,

$A \cdot B + C \cdot D = C \cdot D + A \cdot B$ .

By substitution,

$A \cdot B + C \cdot D = 66$ .

Assume Statement 2 alone. By the commutative property of multiplication,

$A \cdot B = B \cdot A$ and $C \cdot D = D \cdot C$

By two substitutions,

$A \cdot B + C \cdot D = B \cdot A + D \cdot C$

$A \cdot B + C \cdot D = 66$

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Question

Evaluate .

Statement 1:

Statement 2:

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Answer

By the distributive property of multiplication over addition,

$A (B + C) = A \cdot B + A \cdot C$

Assume Statement 1 alone. $A \cdot B = 128$ , so, by substitution,

$A (B + C) = 128+ A \cdot C$ .

But, without knowing , , or their product, it is impossible to determine the value of the expression. Statement 2 provides insufficient information, for similar reasons.

Now assume both statements to be true. Then

$A (B + C) = A \cdot B + A \cdot C$

Since $A \cdot B = 128$ and $A \cdot C = - 37$ , by substitution,

.

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Question

Evaluate $A \cdot B + C \cdot D$ .

Statement 1: $A \cdot B + D \cdot C = 100$

Statement 2: $A \cdot B = C \cdot D$

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Answer

Assume Statmemt 1 alone. By the commutative property of multiplication:

$C \cdot D = D \cdot C$

By the addition property of equality,

$A \cdot B + C \cdot D = A \cdot B + D \cdot C$

By substitution,

$A \cdot B + C \cdot D = 100$

Statement 2 alone, however, does not answer the question. If $A \cdot B = C \cdot D$ , then by substitution,

$A \cdot B + C \cdot D = A \cdot B + A \cdot B = 2 \cdot A \cdot B$ ;

however, without further information, we cannot evaluate this expression.

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