Operations on fractions and Decimals

Question 1

As you can recall, similar fractions are fractions that have the same denominator. The rules on adding and subtracting similar fractions are the same. To add or subtract similar fractions, follow these steps:

Answer 1

Add or subtract the numerators of the given fractions and use the sum or difference as the numerator of the resulting fraction.
Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.
Reduce the answer to its lowest terms, if possible.
To summarize: In order to add or subtract similar fractions, you first need to add or subtract the numerator, then copy the denominator. Afterward, simplify your answer to its lowest terms.

Question 2

Example 1: 3⁄5 + 1⁄5.

Answer 2

Solution:

Step 1: Add the numerators of the given fractions and use the sum or difference as the numerator of the resulting fraction.

operations on fractions and decimals 1
We just add the numerators of the given fractions which are 3 and 1 and put the answer as the numerator of the resulting fraction.

Step 2: Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.

operations on fractions and decimals 2
The denominator of the given fractions is 5. Hence, we will use 5 as the denominator of the common fraction.

Step 3: Reduce the answer to its lowest terms, if possible.

4⁄5 is a fraction in the lowest terms. Hence, no need to simplify it further. Therefore, our final answer is 4⁄5.

Question 3

Example 2: Add 1⁄4 and 2⁄4.

Answer 3

3/4

Question 4

Example 3: Subtract 3⁄21 from 10⁄21.

Answer 4

1/3

Question 5

xample 4: Bea ate 2⁄8 of the pie that her mother prepared. Meanwhile, Bea’s brother ate 4⁄8 of the same pie that Bea ate. What is the total fraction of the pie eaten by Bea and her brother?

Answer 5

Solution:

We can answer this question by adding the fraction of the pie eaten by Bea and the fraction of the pie eaten by her brother. Since 2⁄8 and 4⁄8 are similar fractions, we can use the steps we have for adding similar fractions.

Step 1: Add the numerators of the given fractions and use the sum as the numerator of the resulting fraction.

operations on fractions and decimals 5
Step 2: Copy the denominator of the given fractions and use it as the denominator of the resulting fraction.

operations on fractions and decimals 6
Step 3: Reduce the answer to its lowest terms, if possible.

6⁄8 is not in its lowest terms yet since 6 and 8 have a Greatest Common Factor (GCF) of 2. Hence, we divide both 6 and 8 by 2.

operations on fractions and decimals 7
Therefore, Bea and her brother ate 6⁄8 or ¾ of the pie.

Question 6

These are the steps on how to transform dissimilar fractions into similar fractions:

Answer 6

Find the Least Common Multiple (LCM) of the denominators. The number that you will obtain is the Least Common Denominator (LCD). Use the LCD as the new denominator of the fractions.
Divide the LCM you have obtained by the denominator of the first fraction. Multiply the resulting number by the numerator. The number that you will obtain is the numerator for the new fraction.
Apply Step 2 for the second fraction.

Question 7

Example: Transform the fractions 3⁄5 and 1⁄3 into similar fractions.

Answer 7

Solution:

Let us apply all the steps previously discussed.

Step 1: Find the Least Common Multiple (LCM) of the denominators. The number that you will obtain is the Least Common Denominator (LCD). Use the LCD as the new denominator of the fractions.

The Least Common Multiple of 5 and 3 is 15 (we colored it with purple in the list). 15 will be our Least Common Denominator (LCD).

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We will use 15 as the denominator of our fractions. We leave the numerators of the fractions blank because we need to compute them in the next step.

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Step 2: Divide the LCM you have obtained by the denominator of the first fraction. Multiply the resulting number by the numerator. The number that you will obtain is the numerator for the new fraction.

Let us apply this step to 3⁄5.

operations on fractions and decimals 10

The LCD we have obtained is 15. We divide the LCD by the denominator of ⅗. Thus, 15 ÷ 5 = 3. Afterward, we multiply 3 to the numerator of ⅗. Hence, 3 x 3 = 9. Therefore, the new numerator is 9.

Step 3: Apply Step 2 for the second fraction.

We will do the same thing we performed on ⅗ for the second fraction which is ⅓. We divide the LCD of 15 by the denominator of ⅓ which is 3. Thus, 15 ÷ 3 = 5. Afterward, we multiply 5 by the numerator of ⅓. Hence, 5 x 1 = 5. The new numerator for the second fraction is 5.

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When we transform the fractions 3⁄5 and 1⁄3 into similar fractions, we have 9⁄15 and 5⁄15

Transforming dissimilar fractions into similar fractions is an important step in adding and subtracting dissimilar fractions. This means you should master the method presented above before proceeding to the next section of this reviewer.

Question 8

Here are the steps on how to add or subtract dissimilar fractions:

Answer 8

Change the given dissimilar fractions into similar fractions (refer to the section above for the steps on transforming dissimilar fractions to similar fractions).
Proceed with the steps on addition or subtraction of similar fractions.
Reduce the resulting fraction to its lowest terms, if possible.

Question 9

Example 2: Compute for 1⁄3 – 1⁄4.

Answer 9

olution:

operations on fractions and decimals 13
The LCD of 3 and 4 is 12. Thus, we used it as the new denominator of the fractions. Afterward, we applied the steps on transforming dissimilar fractions into similar fractions. Thus, we obtained 4⁄12 and 3⁄12. In Step 2, we just subtracted the numerators: 4 – 3 = 1 and then copied the denominator of 12. Thus, we obtained a fraction of 1⁄12.

Since 1⁄12 is already in its lowest terms, there is no need to simplify it further. Therefore, the final answer is 1⁄12.

Question 10

Here are the steps you need to follow if you are adding or subtracting mixed numbers:

Answer 10

Add or subtract the whole numbers. The resulting number is the whole number part of the sum or difference.
Add or subtract the proper fractions. If the given fractions are similar fractions, just add or subtract the numerators then copy the denominator. If the given fractions are dissimilar fractions, make the fractions similar first.
Combine the whole number you obtained from Step 1 and the proper fraction you obtained from Step 2 to arrive at a mixed number.
Reduce the proper fraction to its lowest terms, if possible.

Question 11

Example: Add 1 1⁄3 and 4 2⁄5.

Answer 11

Solution:

Step 1: Add the whole numbers. The resulting number is the whole number part of the sum.

The whole number parts of 1 1⁄3 and 4 2⁄5 are 1 and 4, respectively. Adding the whole numbers:

1 + 4 = 5

Therefore, 5 is the whole number part of our sum.

Step 2: Add the proper fractions. If the given fractions are similar fractions, just add the numerators then copy the denominator. If the given fractions are dissimilar fractions, make the fractions similar first.

The proper fractions of 1 1⁄3 and 4 2⁄5 are ⅓ and ⅖, respectively. These proper fractions are dissimilar fractions so we need to transform them first into similar fractions.

If we transform ⅓ and ⅖ into similar fractions, we will have (refer to our previous section to review how to transform dissimilar fractions into similar fractions):

1⁄3 → 5⁄15

2⁄5 → 6⁄15

Now, we add the similar fractions:

5⁄15 + 6⁄15 = 11⁄15

Step 3: Combine the whole number you obtained from Step 1 and the proper fraction you obtained from Step 2 to arrive at a mixed number.

The whole number that we have obtained from Step 1 is 5. Meanwhile, the proper fraction we have obtained from Step 2 is 11⁄15. Combining them, we have 5 11⁄15.

Step 4: Reduce the proper fraction to its lowest terms, if possible.

Since 11⁄15 is in its lowest terms, then we do not need to simplify it.

Therefore, 1 1⁄3 + 4 2⁄5 = 5 11⁄15.

Question 12

Multiplying fractions is a lot easier than adding or subtracting fractions because you do not have to consider whether the fractions are similar or dissimilar. To multiply fractions, all you have to do is follow these three steps:

Answer 12

Multiply the numerators of the given fractions. The resulting number is the numerator of the product (or answer).
Multiply the denominators of the given fractions. The resulting number is the denominator of the product (or answer).
Reduce the product (or answer) to its lowest terms, if possible.
We can summarize these three steps this way: Multiply numerator by numerator and then denominator by denominator. Afterward, reduce the product to its lowest terms.

Question 13

Example 1: Multiply 3⁄4 by 1⁄5.

Answer 13

Solution:

Step 1: Multiply the numerators of the given fractions. The resulting number is the numerator of the product (or answer).

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The numerators of the given fractions are 3 and 1. When we multiply them, we will obtain 3 x 1 = 3. Hence, 3 is the numerator of our resulting fraction.

Step 2: Multiply the denominators of the given fractions. The resulting number is the denominator of the product (or answer).

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Step 3: Reduce the product (or answer) to its lowest terms, if possible.

3⁄20 is a fraction that is already in the lowest terms. Hence, no need to simplify it further.

Therefore, our final answer is 3⁄20.

Let us have more examples:

Example 2: Multiply 5⁄9 by 2⁄4.

Solution:

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Therefore, the product is 5⁄18.

Question 14

Example 3: What is 2⁄5 of 50?

Answer 14

Solution:

The word “of” is actually a signal word for the multiplication of fractions. Hence, the question above can be interpreted also as 2⁄5 × 50.

But, how do we multiply a fraction by a whole number or vice versa?

The answer is simple! Just put a denominator of 1 for the whole number:

2⁄5 × 50⁄1

Afterward, proceed with the steps on multiplying fractions.

2⁄5 × 50⁄1 = 100⁄5

Note that we can simplify 100⁄5 as 20⁄1.

If the denominator of a fraction is 1, it means that the fraction is equal to the whole number indicated in the numerator.

Therefore, 20⁄1 = 20

Hence, 2⁄5 of 50 is equal to 20.

Question 15

Example 4: What is 3⁄4 of 100?

Answer 15

Solution:

This question can be solved using the same method we used for the previous example. Again, the word “of” is a signal word for the multiplication of fractions.

Let us start by putting a denominator of 1 for 100:

3⁄4 × 100⁄1

Multiply the numerators as well as the denominators:

       3⁄4 × 100⁄1 = 300⁄4

We can simplify 300⁄4 as 75⁄1 which is equal to 75.

Hence, 3⁄4 of 100 is equal to 75.

Question 16

Multiplying Fractions Through Cancellation Method.

Answer 16

We can actually make the process of multiplying fractions quicker through the cancellation method. In this method, we “cancel” numbers that have common factors so we can arrive at the product which is already in its lowest terms.

Question 17

Example 1: Multiply 4⁄20 by 5⁄8.

Answer 17

Using the cancellation method, the answer is 1⁄8.

Question 18

Example 2: What is 3⁄7 of 49?

Answer 18

Solution:

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Again, the word “of” is a signal word for the multiplication of fractions.

Therefore, 3⁄7 of 49 is equal to 21.

Question 19

the fraction that when multiplied by the original fraction, the result is 1.

Answer 19

Reciprocal or Multiplicative Inverse of a Number

Question 20

Let’s use fraction 5⁄6 as an example. The reciprocal of this fraction can be obtained by interchanging the positions of the numerator and the denominator. Therefore, the reciprocal of 5⁄6 is simply 6⁄5.

Easy, right? Now, can you determine the reciprocal of the following:

4⁄5, 5⁄8, and 25.

Answer 20

Here are the answers:

The reciprocal of 4⁄5 is 5⁄4.

The reciprocal of 5⁄8 is 8⁄5.

Meanwhile, the reciprocal of 25 is 25⁄1.

Question 21

here are the steps you need to follow so you will be able to divide fractions:

Answer 21

Transform the second fraction (the divisor) into its reciprocal (turn the fraction upside down).
Multiply the first fraction by the reciprocal of the second fraction.
Reduce the obtained fraction to its lowest terms, if possible.

Question 22

Example 1: What is 5⁄6 divided by 6⁄4?

Answer 22

Solution:

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Therefore, 5⁄6 ÷ 6⁄4 = 5⁄9

Question 23

Example 2: Divide 3⁄7 by 1⁄2.

Answer 23

Solution:

Step 1: Transform the second fraction (the divisor) into its reciprocal (turn the fraction upside down).

The second fraction (the divisor) is 1⁄2. Its reciprocal is 2⁄1.

Step 2: Multiply the first fraction by the reciprocal of the second fraction.

3⁄7 × 2⁄1 = 6⁄7

Step 3: Reduce the obtained fraction to its lowest terms, if possible.

6⁄7 is already in its lowest terms. Hence, we do not need to simplify it.

Therefore, 3⁄7 ÷ 1⁄2 = 6⁄7

Question 24

Example 1: Multiply 1 2⁄3 by 2⁄5.

Answer 24

Solution:

The first thing you have to do is to transform the given mixed number into an improper fraction.

1 2⁄3 is a mixed number. If you transform it into an improper fraction, you have 5⁄3.

Afterward, you may now proceed with multiplying 5⁄3 by 2⁄5.

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Lastly, we can reduce 10⁄15 into its lowest terms:

operations on fractions and decimals 21
Therefore, 1 2⁄3 × 2⁄5 = 2⁄3

Question 25

Example 2: Divide 8 3⁄5 by 9.

Answer 25

Solution:

Start by transforming the given mixed number into an improper fraction.

8 3⁄5 = 43⁄5

Now, let’s proceed to divide 43⁄5 by 9. The reciprocal of 9 is 1⁄9.

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Therefore, 8 3⁄5 ÷ 9 = 43⁄45

Question 26

To add decimal numbers, follow these steps:

Answer 26

Align the decimal numbers vertically, with the decimal points lined up.
Add zeros at the end of some decimal numbers so that the decimals will be of the same length.
Add or subtract the digits and put the decimal point in the final answer.

Question 27

Example 1: Delly bought a pencil worth PHP 8.25 and an eraser worth PHP 4.105. How much is the total amount of items that Delly bought?

Answer 27

Solution: We can answer this problem by adding the given amounts which are decimal numbers.

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To solve this problem, we started by aligning the given decimal numbers. Afterward, we added a zero at the end of 8.25 so that it will be of the same length as 4.105. Lastly, we performed column addition from right to left (just like with whole numbers) and put the decimal point by bringing it down.

Therefore, 8.25 + 4.105 = 12.355

Question 28

Example 2: Letty loves jogging. On Monday, she jogged a distance of 3.258 km. Meanwhile, on Tuesday, she jogged a distance of 4.15 km. What is the total distance covered by Letty on Monday and Tuesday?

Answer 28

Solution:

We can answer this problem by adding the given distances which are decimal numbers.

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Therefore, Letty covered a total distance of 7. 408 km on Monday and Tuesday.

Question 29

Example 3: Berto has 2.598 liters of alcohol. He used 0.52 liters for disinfecting his furniture. How many liters of alcohol were left?

Answer 29

Solution:

We can solve this problem by subtracting 0.52 from 2.598

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Therefore, 2.078 liters of alcohol were left.

Question 30

Example 4: What is the difference between 9.453 and 7.38?

Answer 30

Thus, the difference between 9.453 and 7.38 is 2.073.

Question 31

Example 1: Multiply 5.45 by 1.2

Answer 31

Solution:

We start our calculation by ignoring the decimal point and multiplying the numbers just like whole numbers.

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We have obtained 6540 from Step 1 but it is not the final answer yet. We need to put the decimal point somewhere in its digits.

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5.45 has two digits at the right of its decimal point. Thus, it has two decimal places. Meanwhile, 1.2 has one digit at the right of its decimal point. Therefore, it has one decimal place. The total number of decimal places we now have is three (two from 5.45 and one from 1.2). Thus, the final answer must have three decimal places.

To determine where we should put our decimal point in 6540, count three digits from the right then put the decimal point. Hence, the decimal point should be at 6.540

Thus, the answer is 6.540 or 6.54

Question 32

To divide decimal numbers, you may follow these steps:

Answer 32

Move the decimal point of the divisor (the second decimal) to the right until it becomes a whole number.
Move the decimal point in the dividend (the first decimal) to the right the same number of places you move the decimal point in the divisor.
Divide normally just like whole numbers using the new decimals obtained from Step 1 and 2 and put the decimal point to the final answer.
Let us apply these steps to our example below:

Question 33

Example: Divide 32.95 by 0.5

Answer 33

Solution:

Step 1: Move the decimal point of the divisor (the second decimal) to the right until it becomes a whole number.

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We can move one decimal place to the right of 0.5 so that it becomes a whole number (which is 5).

Step 2: Move the decimal point in the dividend (the first decimal) to the right the same number of places you move the decimal point in the divisor.

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Step 3: Divide normally just like whole numbers using the new decimals obtained from Steps 1 and 2 and put the decimal point to the final answer.

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Therefore, 32.95 ÷ 0.5 = 65.9

Question 34

1) Add:
2
3 +
1
4
a)
14
15
b)
13
14
c)
11
15
d)
11
12

Answer 34

D

Question 35

2) Multiply:
3
5 ×
4
6
a)
2
5
b)
1
3
c)
3
8
d) None of the above

Answer 35

A

Question 36

3) Subtract: 2
3
5 − 1
1
5
a) 1
1
5
b) 1
2
5
c) ¾
d) ⅕

Answer 36

B

Question 37

4) Multiply 0.123 by 0.12
a) 0.01476
b) 0.02316
c) 0.00816
d) 0.01216

Answer 37

A

Question 38

5) Divide by 9
1
3
a) 27
b)
1
27
c) 9
d)
1
9

Answer 38

B