How to divide variables

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ISEE Middle Level Quantitative Reasoning › How to divide variables

Questions 1 - 10

is the multiplicative inverse of . Which is the greater quantity?

(a) $\frac{a}{b}$

(b)

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Explanation

is the multiplicative inverse of , so, by definition, . We show that this is insufficient information, however, with two cases.

Case 1: $a = 2, b= \frac{1}{2}$

$ab = 2 \cdot \frac{1}{2} = 1$

$\frac{a}{b} = a \div b = 2 \div \frac{1}{2} = 2 \cdot 2 = 4 > 1$

Case 2: $a= \frac{1}{2}, b = 2$

$ab = \frac{1}{2} \cdot 2 = 1$

$\frac{a}{b} = a \div b = \frac{1}{2} \div 2 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} < 1$

Therefore,it is not clear which, if either, is the greater, $\frac{a}{b}$ or 1.

Suppose you know the values of , , and , and you want to evaluate the expression below.

$A + B^{2}\div C$

Which of the following is the first step you must complete?

Evaluate .

Evaluate $B\div C$ .

Evaluate .

Evaluate $A\div C$ .

Explanation

Use the order of operations, PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

In our expression, there are no parentheses, so square first.

Simplify:

$\frac{5a^{-2}x}{ax^2}$

$\frac{5}{a^3x}$

$\frac{1}{5a^3x}$

$\frac{a}{25a^2x}$

$\frac{x^2}{25a}$

$\frac{5a}{x}$

Explanation

The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:

$\frac{5a^{-2}x}{ax^2} = \frac{5x}{a^2ax^2}$

Now, since all of the variables are positive powers, this is easy. Normally, you would begin by reducing the numeric coefficient. This is not necessary since there is only a 5 in the numerator. Therefore, combine the like variables first:

$\frac{5x}{a^2ax^2} = \frac{5x}{a^3x^2}$

Next, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{5x}{a^3x^2} = \frac{5}{a^3x}$

Jack can paint 3 vases in an hour. Jenny can paint 2 vases in an hour. If they work together, how long will it take them to paint 30 vases?

$6\ \text{hours}$

$3\ \text{hours}$

$9\ \text{hours}$

$10\ \text{hours}$

$15\ \text{hours}$

Explanation

If Jack and Jenny paint the vases together, they will paint 5 vases per hour since Jack's rate of painting is 3 vases per hour, while Jenny's rate is 2 vases per hour.

If there are 30 vases to paint, and they paint 5 vases per hour, it will take 6 hours to paint all the vases because 30 divided by 5 is 6.

$30\div5=6$

$\frac{9x^{4}y^{6}}{3x^{2}y^{2}} =$

$3x^{2}y^{4}$

$6x^{2}y^{4}$ $3x^{2}y^{3}$

$6x^{8}y^{12}$

Explanation

To solve $\frac{9x^{4}y^{6}}{3x^{2}y^{2}}$ separate each part of the terms and divide. When dividing variables with exponents, subtract the exponents.

$9\div 3 = 3$

$\frac{x^{4}}{x^{2}}=x^{4-2} =x^{2}$

$\frac{y^{6}}{y^{2}}=y^{6-2} =y^{4}$

$3x^{2}y^{4}$

Simplify the following expression:

$\frac{t^4x^5p^{26}}{t^2x^4p^{20}}$

$t^2x^{10}p^6$

$\frac{1}{t^2xp^6}$

Explanation

Simplify the following expression:

$\frac{t^4x^5p^{26}}{t^2x^4p^{20}}$

When we are dividing variables, we can simpligy by subtracting the exponent of the variable on bottom from the variable on the top.

So because t has an exponent of 4 on top and 2 on the bottom, the new exponent will be:

$t^4-t^2=t^{4-2}=t^2$

Do the same for our other two exponents (x and p) to get the following

$\frac{t^4x^5p^{26}}{t^2x^4p^{20}}=t^2xp^6$

Making our answer:

Simplify:

$\frac{35x^{-3}y^2z^{-1}}{105ax^2y^{-2}{}z^{-5}}$

$\frac{y^4z^4}{3ax^5}$

$\frac{y^2z^4}{35ax^5z^1}$

$\frac{3y^4z^4}{35ax^5}$

$\frac{xy}{3az^4}$

$\frac{xyz^4}{3a}$

Explanation

The easiest way to begin with questions like this one is to "flip" the negative exponents to the top or bottom of the fraction. When you do this, you make the exponent's sign positive:

$\frac{35x^{-3}y^2z^{-1}}{105ax^2y^{-2}z^{-5}} = \frac{35y^2y^2z^5}{105ax^3x^2z^1}$

Next, go ahead and reduce the numeric coefficient:

$\frac{35y^2y^2z^5}{105ax^3x^2z^1} = \frac{1y^2y^2z^5}{3ax^3x^2z^1}$

Then, combine the like variables first:

$\frac{1y^2y^2z^5}{3ax^3x^2z^1} = \frac{1y^4z^5}{3ax^5z^1}$

Finally, cancel out the variables. Subtract the smaller power from the larger one. Remember that if there is no power listed, it is 1:

$\frac{1y^4z^5}{3ax^5z^1} = \frac{1y^4z^4}{3ax^5} = \frac{y^4z^4}{3ax^5}$

Which of the following is not a factor of 56?

All of these are factors of 56

Explanation

Factors are multiplied together to reach a specific target number. When you divide a target number by one of its factors, the result is always a whole number.

In this case, we can evaluate the numbers that can be multiplied to get 56.

$7\times8=56$