How to Calculate Faster using Cancellation Part 2

In the previous post, we have learned how to use cancellation to reduce fractions to lowest terms and fractions. In this video, we are going to use cancellation in other calculations.

Multiplying Fractions by Whole Numbers

Example 1: Calculate 4 \times \dfrac{3}{2}.

Solution/Explanation

As we have learned before, we can place 1 on the denominators of whole numbers. Therefore,

4 \times \dfrac{3}{2}

can be written as

\dfrac{4}{1} \times \dfrac{3}{2}.

From here, we can cancel 4 and 2 by dividing both of them by 2.

cancellation

This gives us

\dfrac{2}{1}\times \dfrac{3}{1} = \dfrac{6}{1} = 6.

Dividing Algebraic Expressions

Example 2: \dfrac{12m^3n^4}{3mn^3}

Solution/Explanation

From the expression, we can cancel 12 and 3 by dividing both the numerator and denominator by 3. This gives us 4/1. Next, we can divide m^3 by m, where one m  can be cancelled. This leaves m^2 in the numerator.

cancellation 2

Now, n^4 can be written as (n^3)(n) and n^3 can be cancelled. This leaves n in the numerator. Therefore, the final answer is 4m^2n

Solving Equations

Example 3: \dfrac{3}{4}x + \frac{2}{3}= 8

Multiplying everything by 12, the least common multiple of 4 and 3, we have

12(\dfrac{3}{4}x) + 12 (\dfrac{2}{3}) = 12(8)

We can cancel out 12 and 4 in the first term by dividing by 4. This leaves us 3(3)x = 9x in the numerator. In the second term, we can cancel out 12 and 3, which leaves 4(2) = 8 in the numerator. The right hand side becomes 96. The resulting equation is

9x + 8 = 96

If we want to solve the equation, we have

9x = 96 - 8

9x = 88

x = \frac{88}{9}

As we can see, cancellation is very useful in simplifying calculations. First it speeds up calculations and second it lessens the probability of computational errors because the numbers get smaller.

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