Browse Tag: cancellation

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.

How to Calculate Faster using Cancellation Part 1

Cancellation is one of the great techniques in making calculations faster. This technique is used in simplifying fractions, rational expressions, and equations in Algebra. In this post, we are going to learn some of the cancellation techniques that are usually not in schools but can be a helpful strategy in taking examinations like the Civil Service Exam.

1. Getting the Lowest Terms of a Fraction

Cancellation can be used to simplify fractions in order to convert them to lowest terms. In general, in order to simplify fractions, we have to get the greatest common denominator of the numerator and the denominator; however, making use of cancellation several times until the fraction is in lowest terms is also a good strategy especially for large numbers .

cancellation

 

In the first example above, cancellation is used to simplify 6/9 to 2/3 by dividing both the numerator and denominator by 3.  In the second example, cancellation was used twice: first, 24/32 is divided by 4 to obtain 6/8, and then was divided by 2 to obtain 3/4.

2. Multiplying Fractions

Cancellation can also be used to simplify multiplication of fractions. You can cancel any pair of number where one is on the numerator and the other is on the denominator.

cancellation

In the example above, 4/9 is multiplied by 3/16.

(1) We can cancel out 3 and 9 by dividing both of them by 3. We get 1 in the numerator and 3 in the denominator.

(2) We can also cancel out 4 and 16 by dividing both of them by 4. This gives us 1 and 4 respectively.

cancellation
Click image to enlarge

In the last example, we have

\dfrac{7}{15} \times \dfrac{9}{8} \times \dfrac{4}{21}.

(1)  9 and 15 are cancelled by dividing both of them by 3. This results to 3 and 5, respectively.
(2) 4 and 8 are cancelled out by dividing both of them by 4. This results to 1 and 2, respectively.
(3) 7 and 21 are cancelled out by dividing both of them by 7. This results to 1 and 3, respectively.
(4) 3 and 3 are cancelled out by dividing both of them by 3. The result is 1.

This results to the simplified fraction to

\dfrac{1}{5} \times \dfrac{1}{2} \times \dfrac{1}{1} = \dfrac{1}{10}.

The examples above are discussed in the following video. The language used is mixed Tagalog and English.

In the next post, we will discuss more about cancellation.