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ISEE Middle Level Quantitative : Algebraic Concepts

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All ISEE Middle Level Quantitative Resources

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Example Questions

Example Question #4 : How To Find The Solution To An Equation

Choose the best answer from the four choices given.

Richard reaches into his pocket and determines that he has $3.85 in quarters and nickels. When he pulls out his fistful of change, though, he is dismayed to realize that what he thought were quarters are really pennies and what he thought to be nickels are really dimes. If he has 12 pennies, how much is his change worth?

Possible Answers:

$\$1.82$

$\$3.12$

$\$1.32$

$\$3.02$

Correct answer:

$\$1.82$

Explanation:

If Richard has 12 pennies, that means that he thought he had 12 quarters ($3.00) and 85 cents worth of nickels (17 nickels). Since what he thought were nickels are really dimes, it means that he has $1.70 worth of dimes to go along with his 12 cents in pennies.

$\$1.70+\$0.12=\$1.82$

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Example Question #4 : How To Find The Solution To An Equation

Using the information given in each question, compare the quantity in Column A to the quantity in Column B.

5y-9x=10

Column A Column B

the slope of the y-intercept

this equation of this equation

Possible Answers:

The quantity in Column B is greater.

The two quantities are equal.

The relationship cannot be determined from the information given.

The quantity in Column A is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

Divide each of the elements of the equation by 5 to get it into y=mx+b form.

$\frac{5y}{5}-\frac{9x}{5}=\frac{10}{5}$

$y-\frac{9}{5}x=2$

$y=\frac{9}{5}x+2$

Thus, the slope is 1.8 and the y-intercept is 2, so B is the correct answer.

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Example Question #3 : How To Find The Solution To An Equation

x(3x-7)= 40

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

Rewrite in standard form:

x(3x-7)= 40

$x \cdot 3x-x \cdot7 = 40$

$3x^{2}-7x = 40$

$3x^{2}-7x- 40 = 40 - 40$

$3x^{2}-7x- 40 = 0$

Use the -method to factor the quadratic expression, splitting the middle term using integers whose product is 3 (-40)= -120 and whose sum is . These two numbers are -15,8 , so the equation becomes:

$3x^{2}-15x+8x- 40 = 0$

$\left ( 3x^{2}-15x \right ) +\left ( 8x- 40 \right ) = 0$

$3x \left ( x-5 \right ) + 8 \left ( x-5 \right ) = 0$

$\left (3x+ 8 \right ) \left ( x-5 \right ) = 0$

Set each linear binomial to 0 and solve:

3x+ 8 = 0

3x+ 8 -8= 0 -8

3x= -8

$3x\div 3= -8 \div 3$

$x = -2 \frac{2}{3}$

x - 5 = 0

x - 5+ 5 = 0 + 5

x = 5

Therefore, it is not clear whether is greater than or less than 3.

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Example Question #5 : How To Find The Solution To An Equation

$\left ( t- 10\right ) (2t-3) = -35$

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

Correct answer:

(a) is greater

Explanation:

Rewrite in standard form:

$\left ( t- 10\right ) (2t-3) = -35$

$t \cdot 2t -t \cdot 3 -10 \cdot 2t + 10 \cdot 3 = -35$

$2t^{2} -3t -20 t + 30 = -35$

$2t^{2} -23t + 30 = -35$

$2t^{2} -23t + 30+ 35 = -35 + 35$

$2t^{2} -23t + 65 =0$

Use the -method to factor the quadratic expression, splitting the middle term using integers whose product is $2 \times65=130$ and whose sum is . These integers are -13,-10 , so the equation becomes:

$2t^{2} -10t - 13t + 65 =0$

Group and solve:

$\left ( 2t^{2} -10t \right )-\left ( 13t - 65 \right ) =0$

$2t \left ( t-5\right )-13\left ( t-5 \right ) =0$

$\left ( 2t -13 \right )\left ( t-5\right ) =0$

Set each linear binomial to 0 and solve:

2t-13 = 0

2t-13 +13= 0+13

2t = 13

$2t \div 2 = 13 \div 2$

$t = 6 \frac{1}{2}$

t - 5 = 0

t - 5+ 5 = 0 + 5

t = 5

In either case, t > 0 .

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Example Question #8 : Equations

x(x+1)= 30

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

Rewrite in standard form:

x(x+1)= 30

$x \cdot x+x \cdot 1 = 30$

$x ^{2}+x = 30$

$x ^{2}+x- 30 = 30 - 30$

$x ^{2}+x- 30 = 0$

Factor the quadratic expression as (x + ?)(x + ?) , replacing the question marks with two integers whose product is and whose sum is 1. These integers are -5,6 , so the equation becomes:

(x-5)(x+6)= 0

Set each linear binomial to 0 and solve:

$x-5= 0 \Rightarrow x= 5$

$x+ 6 = 0 \Rightarrow x = -6$

Therefore, it is not clear whether is greater than or less than 0.

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Example Question #4 : How To Find The Solution To An Equation

y (y + 7) = 30

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

Correct answer:

(b) is greater

Explanation:

Rewrite in standard form:

y (y + 7) = 30

$y \cdot y +y \cdot 7 = 30$

$y ^{2} +7y = 30$

$y ^{2} +7y- 30 = 30 - 30$

$y ^{2} +7y- 30 = 0$

Factor the quadratic expression as (y + ?)(y + ?) , replacing the question marks with two integers whose product is and whose sum is 7. These integers are -3,10 , so rewrite:

(y -3)(y+ 10) = 0

Set each linear binomial to 0 and solve:

$y- 3 \Rightarrow y= 3$

$y+10= 0 \Rightarrow y = -10$

Both solutions are less than 5.

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Example Question #5 : How To Find The Solution To An Equation

$y^{2} + 50 = 15y$

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(a) is greater

It is impossible to tell from the information given

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

Rewrite in standard form:

$y^{2} + 50 = 15y$

$y^{2} + 50 -15y= 15y -15y$

$y^{2} -15y+ 50= 0$

Factor the quadratic expression as (y + ?)(y + ?) , replacing the question marks with two integers whose product is 50 and whose sum is . These integers are -5,-10 , so rewrite the equation as:

(y -5)(y-10) = 0

Set each linear binomial to 0 and solve:

$y -5= 0\Rightarrow y = 5$

$y -10= 0\Rightarrow y = 10$

It is unclear whether is equal to or greater than 5.

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Example Question #11 : Equations

$\left (x-5 \right )^{2} = 36$

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

Use the square root method to solve this equation:

$\left (x-5 \right )^{2} = 36$

$x-5 =- 6 \textrm{ or }x-5 = 6$

Solve each equation separately:

x-5 =- 6

x-5 +5=- 6 + 5

x = -1

x-5 = 6

x-5 +5= 6+5

x = 11

In either case, x > - 6

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Example Question #11 : How To Find The Solution To An Equation

$\frac{x}{3}-9=11$

5y+5=100

Column A Column B
x y

Possible Answers:

The quantity in Column B is greater.

There can be no relationship determined from the two quantities.

The quantities are equal.

The quantity in Column A is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

First, off you need to find the quantities of x and y. For the first equation, add 9 to both sides so that $\frac{x}{3}=20.$ Multiply both sides by 3 so that x=60. Then, solve the second equation. First, subtract 5 from both sides so that 5y=95 . Divide both sides by 5 so that y=19. Therefore, the quantity in Column A is greater.

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Example Question #11 : How To Find The Solution To An Equation

$\frac{x}{2}+7=19$

3y-14=7

Column A Column B

x y

Possible Answers:

The quantities in each column are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

There is no way to determine the relationship between the columns.

Correct answer:

The quantity in Column A is greater.

Explanation:

First, you must solve for x and y. To solve for x, first subtract 7 from both sides, so that you get $\frac{x}{2}=12$ . Multiply both sides by 2, giving you 24 for x. To solve for y, first add 14 to both sides, which gives you 3y=21. Divide both sides by 3 to give you 7. Therefore, Column A is greater.

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : Algebraic Concepts

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

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #4 : How To Find The Solution To An Equation

Example Question #4 : How To Find The Solution To An Equation

Example Question #3 : How To Find The Solution To An Equation

Example Question #5 : How To Find The Solution To An Equation

Example Question #8 : Equations

Example Question #4 : How To Find The Solution To An Equation

Example Question #5 : How To Find The Solution To An Equation

Example Question #11 : Equations

Example Question #11 : How To Find The Solution To An Equation

Example Question #11 : How To Find The Solution To An Equation

All ISEE Middle Level Quantitative Resources

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