How to find the solution to an equation - ISEE Lower Level Quantitative Reasoning

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Question

What is the value of x in the equation?

$\dpi{100} \frac{200+50}{5}=x$

Answer

First, add $\dpi{100} 200+50=250$ .

Next, divide $\dpi{100} 250\div 5=50$ .

So $\dpi{100} x=50$

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Question

Simplify.

Answer

When simplifying an expression, you must combine like terms. There are two types of terms in this expression: “x’s” and whole numbers. Combine in two steps:

x’s:

whole numbers:

The simplified expression is: .

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Question

Solve, when .

$-6x^{2}+12x+3=$

Answer

To solve, insert for each :

$-6(-2)^{2}+12(-2)+3=$

Simplify:

*Common error: When solving this part of the equation always remember the order of operations (PEMDAS) and square the number in the () BEFORE multiplying!

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Question

Solve for .

$5x^{2}-14=6$

Answer

$5(-2)^{2}-14=6$

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Question

Use the equations to answer the question.

What is ?

Answer

First, you need to find what would make and true in their respective equations. equals 1 and equals 4. The next step is to add those together, which gives you 5.

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Question

What story best fits the expression $9\times3$ ?

Answer

Nell's story fits best because if she bought 9 bags with 3 pieces each, that would be 27 total. This fits best with the $9\times 3=27$ equation.

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Question

Max and Sarah both drove at the same speed. It took Max 36 minutes to drive 3 miles. How many minutes did it take for Sarah to drive 6 miles?

Answer

Since it took Max 36 minutes to drive 3 miles, it must take double the amount of time to drive 6 miles since they're both driving at the same speed. Thus, 72 is the correct answer.

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Question

What is equal to in this equation:

Answer

Find the number that makes the equation true. 2 works because when plugged in, the left side of the equation becomes 13, making the whole equation true.

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Question

Five more than a number is equal to $\frac{3}{5}$ of twenty-five . What is the number?

Answer

From the question, we know that plus a number equals $\frac{3}{5}$ of . In order to find out what $\frac{3}{5}$ of is, multiply $\frac{3}{5}$ by $\frac{25}{1}$ .

$\frac{3}{5}\times\frac{25}{1}=\frac{75}{5}$ , or .

The number we are looking for needs to be five less than , or .

You can also solve this algebraically by setting up this equation and solving:

$5+x=\frac{3}{5}\times25$

Subtract from both sides of the equation.

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Question

Solve for .

Answer

To solve for , we want to isolate , or get it by itself.

Add to both sides of the equation.

Now we need to divide both sides by the coefficient of , i.e. the number directly in front of the .

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Question

Solve for .

Answer

First, subtract the three from both sides:

$18-3=3x+3\left ( -3\right )$

Then, divide by three on both sides:

$\frac{15}{3}=\frac{3x}{3}$

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Question

Alice has 30 flowers to put in 6 different vases. Which expression represents the number of flowers in each vase?

Answer

Since there are 30 total flowers and 6 vases, we divide 30 by 6. The result is 5 flowers per vase.

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Question

A water bottle, basketball, and whistle all weigh a combined 5 kg. If the water bottle weighs 1 kg and the basketball weighs 3 kg, how much does the whistle weigh?

Answer

Since the total weight is 5 kg and the other two items weigh a total of 4 kg, subtract 4 from 5. Therefore the whistle weighs 1 kg.

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Question

Michael bought 6 bags of candy at a cost of $4 per bag. Which equation represents the amount of money he spent?

Answer

Since he bought 6 bags of candy at $4 each, we need to multiply $6\times 4.$ The result is $24. Therefore, the equation is $6\times 4=24.$

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Question

Ten less than one-half of a number is sixty. What is the number?

Answer

If we let be the number, then "one-half a number" can be written as

$\frac{1}{2}x$

"Ten less than one-half of a number" can be written as

$\frac{1}{2}x - 10$

Set this equal to sixty and solve:

$\frac{1}{2}x - 10 = 60$

By the addition property of equality:

$\frac{1}{2}x - 10+ 10 = 60 + 10$

$\frac{1}{2}x = 70$

By the multiplication property of equality:

$\frac{1}{2}x \cdot 2 = 70 \cdot 2$

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Question

Which of the following phrases can be written as the algebraic expression $\frac{x}{5} -8$ ?

Answer

$\frac{x}{5} -8$ is eight subtracted from $\frac{x}{5}$ .

$\frac{x}{5}$ is the quotient of a number () and five.

We combine these -

$\frac{x}{5} -8$ is eight subtracted from the quotient of a number and five.

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Question

Forty more than one-third of a number is eighty. What is the number?

Answer

If we let be the number, then "one-third of a number" can be written as

$\frac{1}{3}x$ .

"Forty more than one-third of a number" can be written as

$\frac{1}{3}x +40$ .

Set this equal to eighty and solve:

$\frac{1}{3}x +40 = 80$

By the subtraction property of equality:

$\frac{1}{3}x +40 - 40= 80 - 40$

$\frac{1}{3}x = 40$

By the multiplication property of equality:

$3 \cdot \frac{1}{3}x =3 \cdot 40$

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Question

One third of the difference of a number and fifty is ninety-five. What is the number?

Answer

If we let be the number, "the difference of a number and fifty" is

.

"One third of the difference of a number and fifty" can be written as

$\frac {1}{3}\left \( x - 50 \right \)$ .

Set this equal to ninety-five and solve:

$\frac {1}{3}\left \( x - 50 \right \) = 95$

By the multiplication property of equality:

$3 \cdot \frac {1}{3}\left \( x - 50 \right \) = 3 \cdot 95$

By the addition property of equality:

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Question

Which of the following phrases can be written as the algebraic expression $10- \frac{x}{7}$ ?

Answer

$10- \frac{x}{7}$ is $\frac{x }{7}$ subtracted from ten.

$\frac{x }{7}$ is the quotient of a number () and seven.

Therefore,

$10- \frac{x}{7}$ is the quotient of a number and seven subtracted from ten.

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Question

Which of the following phrases can be written as the algebraic expression $\frac{14 -x}{9}$ ?

Answer

$\frac{14 -x}{9}$ is nine divided into .

is the difference of 14 and a number ().

Therefore,

$\frac{14 -x}{9}$ is nine divided into the difference of 14 and a number.

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