How to divide variables - ISEE Middle Level Quantitative Reasoning

(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)

Therefore, 30 is the correct answer.

(The numerator and the denominator are both multiplied by 10 in order to convert the fraction to whole numbers.)

Therefore, 90 is the correct answer.

Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:

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

Since all of the variables are positive powers, this is easy. Start by reducing the numeric coefficient:

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

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:

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:

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

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:

Next, go ahead and reduce the numeric coefficient:

Then, combine the like variables first:

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

To start, notice that the left side of the equation has a common factor of . You can factor this out:

becomes

Next, you can divide both sides by since you know that it does not equal :

Simplifying the right side, you get:

Finally, subtract from both sides:

To start, notice that the left side of the equation has common factors of and . You can factor these out:

becomes...

Next, you can divide both sides by since you know that it does not equal :

Simplify the right side by canceling the and reducing the :

Finally, add to both sides:

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

Case 1:

Case 2:

Therefore,it is not clear which, if either, is the greater, or 1.

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

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

First convert each mixed number into and improper fraction

Then convert the operation to multiplication and flip the second fraction

Reduce where possible and multiply to solve:

The answer is

This problem looks strange since it mostly contains variables, in particular unknown variables. Still remember the order of operations: parentheses, exponents, multiplication, division, addition, and subtraction. We still use those operations in this case. All we must do is move the variables around to solve for .

In this case we can subtract (additive inverse) to the other side first.

Rewrite our problem.

Stop for a moment and look at the progress. Go back to the original question, solve for . Our problem shows about to be multiplied to , but we do not necessarily have to multiply because we can divide by to both sides of the equation, which would save us a step as well as have B by itself.

To simplify, we want to get the same base in both the numerator and the denominator. This will allow us to easily manipulate the exponents.

We can write:

and

Replace the fraction with these values and simplify.

To simplify, we want to get the same base in both the numerator and the denominator. This will allow us to easily manipulate the exponents.

Replace the denominator with this value and simplify.

A trick to find multiples of 9 is that when the sum of the digits in a number is divisible by 9, then the full number is also divisible by 9.

When the digits of 963 are added together, the sum is 18.

18 is divisible by 9, and so the number 963 must also be divisible by 9.

Check your work:

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.

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.

12 is not a factor of 56. When you divide 56 by 12, the answer is not a whole number.

The number 756 can be segmented into 700 and 56. Given that 700 is divisble by 7 (700 divided by 7 is 100) and that 56 is divisible by 7 (56 divided by 7 is 8), 756, as the aggregate of 700 and 56, is also divisble by 7.