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ISEE Upper Level Quantitative : How to find the solution to an equation

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #31 : How To Find The Solution To An Equation

x + 3y = 28

3x + 2y = 35

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) and (b) are equal

Explanation:

Multiply both sides of the top equation by 3, and subtract both sides of the second equation.

x + 3y = 28

$3\left ( x + 3y \right )= 3 \cdot 28$

$3 x + 9y \right )= 84$

$\underline{3x + 2y = 35}$

7y = 49

$7y \div 7= 49 \div 7$

y = 7

Now substitute to find :

x + 3y = 28

$x + 3 \cdot 7 = 28$

x + 21= 28

x + 21-21= 28-21

x = 7

The two are equal.

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Example Question #32 : How To Find The Solution To An Equation

4x +y = 20

2x + 3y = 30

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

Correct answer:

(b) is greater

Explanation:

Triple both sides of the top equation, and subtract both sides of the bottom equation:

4x +y = 20

$3 \left ( 4x +y \right ) = 3\left ( 20\right )$

12x +3y = 60

$\underline{2x + 3y = 30}$

10x =30

$10x \div 10 = 30 \div 10$

x = 3

Now substitute to find :

4x +y = 20

$4 \cdot 3 +y = 20$

12 +y = 20

12 +y -12= 20 -12

y = 8

This makes y > x

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Example Question #33 : How To Find The Solution To An Equation

t = |u - 1|

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We show that it cannot be determined which of and , if either, is greater, by showing one case in which t > u and one case in which t < u .

Case 1: u = 0 . Then

t = |u - 1| = |0 - 1| = | - 1| = 1

and t > u .

Case 2: u = 1 . Then

t = |u - 1| = |1 - 1| = | 0| = 0

and t < u .

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Example Question #34 : How To Find The Solution To An Equation

t = |u + 1|

Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) is greater

Explanation:

If u < 0 , then, since is nonnegative, t > u .

If $u \geq 0$ , then $u + 1 \geq 1 > 0$ , so the equation becomes t = u + 1 . Therefore, t > u .

Either way, (a) is greater.

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Example Question #35 : How To Find The Solution To An Equation

x + y = 10

Which is the greater quantity?

(a) 5y + 5x

(b) 50

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

x + y = 10

By the commutative property,

y + x = 10

By the multiplication property of equality,

$5 \left ( y + x \right )= 5\cdot 10$

By distribution,

5y + 5x = 5 0

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Example Question #36 : How To Find The Solution To An Equation

is a positive number. Which is the greater quantity?

(a) $N ^{2}$

(b) $N ^{3} + N$

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(b) is greater

Explanation:

If N > 1 , then $N ^{3}+ N > N ^{3}= N ^{2} \cdot N > N ^{2} \cdot 1 = N ^{2}$

If N = 1 , then $N ^{3}+ N = 1 ^{3}+1 = 2 > 1 = 1^{2} = N ^{2}$

If 0 < N < 1 , then $N ^{3}+ N > N = N \cdot 1 > N \cdot N = N ^{2}$

Therefore, regardless of the value of , as long as is positive, $N ^{3} + N > N ^{2}$ , and (b) is greater.

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Example Question #37 : How To Find The Solution To An Equation

$100 x + 100 y = \frac{1}{5}$

Which is the greater quantity?

(a) x + y

(b) 0.005

Possible Answers:

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

$100 x + 100 y = \frac{1}{5}$

From the multiplication property of equality,

$\frac{1}{100} \cdot \left ( 100 x + 100 y \right ) = \frac{1}{100} \cdot \frac{1}{5}$ .

By distribution,

$x + y = \frac{1}{500} = 0.002 < 0.005$ .

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Example Question #38 : How To Find The Solution To An Equation

5t = 6u

Which is the greater quantity?

(a) | t |

(b) |u|

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

5t = 6u

$5t \div 5= 6u\div 5$

$t = \frac{6}{5} u$

Take the absolute value of both sides:

$\left |t \right|= \left | \frac{6}{5} u \right |$

$\left |t \right|= \frac{6}{5} \left | u \right | > \left | u \right |$

This makes (a) greater.

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Example Question #39 : How To Find The Solution To An Equation

$4 A + 4 B = \frac{1}{25}$

10 C +10 D = 0.1

Which is the greater quantity?

(a) A + B

(b) C + D

Possible Answers:

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

(a) $4 A + 4 B = \frac{1}{25}$

$\frac{1}{4} \cdot \left ( 4 A + 4 B \right ) = \frac{1}{4} \cdot \frac{1}{25}$

$A + B = \frac{1}{100 }$

(b) 10 C +10 D = 0.1

Rewrite as a fraction:

$10 C +10 D = \frac{1}{10 }$

$\frac{1}{10 } \cdot \left ( 10 C +10 D \right ) = \frac{1}{10 } \cdot \frac{1}{10 }$

$C + D= \frac{1}{100 }$

A + B = C + D

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Example Question #40 : How To Find The Solution To An Equation

Define f(x) as follows:

$f(x) = \left\{\begin{matrix} x -3 \textrm{ if }x > 0 \\ x + 3 \textrm{ if }x \leq 0 \end{matrix}\right.$

Which is the greater quantity?

(a) f(5) + f(-5)

(b) f(0)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

(a) f(5) can be evaluated by using the definition of for positive :

f (x) = x - 3

f (5) =5 - 3 = 2

f(-5) can be evaluated by using the definition of for nonpositive :

f (x) = x + 3

f (-5) =-5 + 3 = -2

Add: f(5) + f(-5) = 2 + (-2) = 0

(b) f(0) can be evaluated by using the definition of for nonpositive :

f (0) = 0 + 3 = 3

f (0) > f(5) + f(-5)

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the solution to an equation

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #31 : How To Find The Solution To An Equation

Example Question #32 : How To Find The Solution To An Equation

Example Question #33 : How To Find The Solution To An Equation

Example Question #34 : How To Find The Solution To An Equation

Example Question #35 : How To Find The Solution To An Equation

Example Question #36 : How To Find The Solution To An Equation

Example Question #37 : How To Find The Solution To An Equation

Example Question #38 : How To Find The Solution To An Equation

Example Question #39 : How To Find The Solution To An Equation

Example Question #40 : How To Find The Solution To An Equation

All ISEE Upper Level Quantitative Resources

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