Examples with solutions for Addition of Fractions: Finding a Common Denominator by Multiplying the Denominators

Exercise #1

Solve the following exercise:

12+29= \frac{1}{2}+\frac{2}{9}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 2 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 2 and 9

In this case, the common denominator is 18

Now we'll multiply each fraction by the appropriate number to reach the denominator 18

We'll multiply the first fraction by 9

We'll multiply the second fraction by 2

1×92×9+2×29×2=918+418 \frac{1\times9}{2\times9}+\frac{2\times2}{9\times2}=\frac{9}{18}+\frac{4}{18}

Now we'll combine and get:

9+418=1318 \frac{9+4}{18}=\frac{13}{18}

Answer

1318 \frac{13}{18}

Exercise #2

Solve the following exercise:

14+19= \frac{1}{4}+\frac{1}{9}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 4 and 9

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 9

In this case, the common denominator is 36

Now we'll multiply each fraction by the appropriate number to reach the denominator 36

We'll multiply the first fraction by 9

We'll multiply the second fraction by 4

1×94×9+1×49×4=936+436 \frac{1\times9}{4\times9}+\frac{1\times4}{9\times4}=\frac{9}{36}+\frac{4}{36}

Now we'll combine and get:

9+436=1336 \frac{9+4}{36}=\frac{13}{36}

Answer

1336 \frac{13}{36}

Exercise #3

Solve the following exercise:

14+26= \frac{1}{4}+\frac{2}{6}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 4 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 4 and 6

In this case, the common denominator is 12

Now we'll multiply each fraction by the appropriate number to reach the denominator 12

We'll multiply the first fraction by 3

We'll multiply the second fraction by 2

1×34×3+2×26×2=312+412 \frac{1\times3}{4\times3}+\frac{2\times2}{6\times2}=\frac{3}{12}+\frac{4}{12}

Now we'll combine and get:

3+412=712 \frac{3+4}{12}=\frac{7}{12}

Answer

712 \frac{7}{12}

Exercise #4

Solve the following exercise:

15+13= \frac{1}{5}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

1×35×3+1×53×5=315+515 \frac{1\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{3}{15}+\frac{5}{15}

Now we'll combine and get:

3+515=815 \frac{3+5}{15}=\frac{8}{15}

Answer

815 \frac{8}{15}

Exercise #5

Solve the following exercise:

17+13= \frac{1}{7}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 7 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 7 and 3

In this case, the common denominator is 21

Now we'll multiply each fraction by the appropriate number to reach the denominator 21

We'll multiply the first fraction by 3

We'll multiply the second fraction by 7

1×37×3+1×73×7=321+721 \frac{1\times3}{7\times3}+\frac{1\times7}{3\times7}=\frac{3}{21}+\frac{7}{21}

Now we'll combine and get:

3+721=1021 \frac{3+7}{21}=\frac{10}{21}

Answer

1021 \frac{10}{21}

Exercise #6

Solve the following exercise:

210+13= \frac{2}{10}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the least common denominator between 10 and 3

To find the least common denominator, we need to find a number that is divisible by both 10 and 3

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 3

We'll multiply the second fraction by 10

2×310×3+1×103×10=630+1030 \frac{2\times3}{10\times3}+\frac{1\times10}{3\times10}=\frac{6}{30}+\frac{10}{30}

Now we'll combine and get:

6+1030=1630 \frac{6+10}{30}=\frac{16}{30}

Answer

1630 \frac{16}{30}

Exercise #7

Solve the following exercise:

25+13= \frac{2}{5}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 5 and 3

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

2×35×3+1×53×5=615+515 \frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}

Now we'll combine and get:

6+515=1115 \frac{6+5}{15}=\frac{11}{15}

Answer

1115 \frac{11}{15}

Exercise #8

Solve the following exercise:

25+16= \frac{2}{5}+\frac{1}{6}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 5 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 6

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

2×65×6+1×56×5=1230+530 \frac{2\times6}{5\times6}+\frac{1\times5}{6\times5}=\frac{12}{30}+\frac{5}{30}

Now we'll combine and get:

12+530=1730 \frac{12+5}{30}=\frac{17}{30}

Answer

1730 \frac{17}{30}

Exercise #9

Solve the following exercise:

25+36= \frac{2}{5}+\frac{3}{6}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 5 and 6

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 6

In this case, the common denominator is 30

Now we'll multiply each fraction by the appropriate number to reach the denominator 30

We'll multiply the first fraction by 6

We'll multiply the second fraction by 5

2×65×6+3×56×5=1230+1530 \frac{2\times6}{5\times6}+\frac{3\times5}{6\times5}=\frac{12}{30}+\frac{15}{30}

Now we'll combine and get:

12+1530=2730 \frac{12+15}{30}=\frac{27}{30}

Answer

2730 \frac{27}{30}

Exercise #10

Solve the following exercise:

38+23= \frac{3}{8}+\frac{2}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the least common multiple (LCM) between 8 and 3

To find the least common multiple, we need to find a number that is divisible by both 8 and 3

In this case, the common multiple is 24

Now we'll multiply each fraction by the appropriate number to reach the denominator 24

We'll multiply the first fraction by 3

We'll multiply the second fraction by 8

3×38×3+2×83×8=924+1624 \frac{3\times3}{8\times3}+\frac{2\times8}{3\times8}=\frac{9}{24}+\frac{16}{24}

Now we'll combine and get:

9+1624=2524 \frac{9+16}{24}=\frac{25}{24}

Answer

2524 \frac{25}{24}

Exercise #11

Solve the following exercise:

25+13= \frac{2}{5}+\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the least common denominator between 5 and 3

To find the least common denominator, we need to find a number that is divisible by both 5 and 3

In this case, the common denominator is 15

Now we'll multiply each fraction by the appropriate number to reach the denominator 15

We'll multiply the first fraction by 3

We'll multiply the second fraction by 5

2×35×3+1×53×5=615+515 \frac{2\times3}{5\times3}+\frac{1\times5}{3\times5}=\frac{6}{15}+\frac{5}{15}

Now we'll combine and get:

6+515=1115 \frac{6+5}{15}=\frac{11}{15}

Answer

1115 \frac{11}{15}

Exercise #12

Solve the following exercise:

25+14= \frac{2}{5}+\frac{1}{4}=

Video Solution

Step-by-Step Solution

Let's try to find the lowest common denominator between 5 and 4

To find the lowest common denominator, we need to find a number that is divisible by both 5 and 4

In this case, the common denominator is 20

Now we'll multiply each fraction by the appropriate number to reach the denominator 20

We'll multiply the first fraction by 4

We'll multiply the second fraction by 5

2×45×4+1×54×5=820+520 \frac{2\times4}{5\times4}+\frac{1\times5}{4\times5}=\frac{8}{20}+\frac{5}{20}

Now we'll combine and get:

8+520=1320 \frac{8+5}{20}=\frac{13}{20}

Answer

1320 \frac{13}{20}

Exercise #13

Solve the following exercise:

28+23= \frac{2}{8}+\frac{2}{3}=

Video Solution

Step-by-Step Solution

Let's try to find the least common multiple (LCM) between 8 and 3

To find the least common multiple, we need to find a number that is divisible by both 8 and 3

In this case, the common multiple is 24

Now we'll multiply each fraction by the appropriate number to reach the denominator 24

We'll multiply the first fraction by 3

We'll multiply the second fraction by 8

2×38×3+2×83×8=624+1624 \frac{2\times3}{8\times3}+\frac{2\times8}{3\times8}=\frac{6}{24}+\frac{16}{24}

Now we'll combine and get:

6+1624=2224 \frac{6+16}{24}=\frac{22}{24}

Answer

2224 \frac{22}{24}

Exercise #14

Solve the following exercise:

110+13=? \frac{1}{10}+\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve the addition of fractions 110+13 \frac{1}{10} + \frac{1}{3} , we must first find a common denominator.

  • Step 1: Find the Least Common Multiple (LCM) of the denominators, 10 and 3. By multiplying these denominators, the LCM is 10×3=30 10 \times 3 = 30 .

  • Step 2: Rewrite each fraction with the common denominator of 30:
    - Convert 110 \frac{1}{10} to an equivalent fraction with a denominator of 30. Multiply both numerator and denominator by 3: 110=1×310×3=330 \frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}
    - Convert 13 \frac{1}{3} to an equivalent fraction with a denominator of 30. Multiply both numerator and denominator by 10: 13=1×103×10=1030 \frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}

  • Step 3: Add the equivalent fractions: 330+1030=3+1030=1330 \frac{3}{30} + \frac{10}{30} = \frac{3 + 10}{30} = \frac{13}{30}

  • Step 4: Simplify the resulting fraction. Since 13 is a prime number and does not divide 30, 1330\frac{13}{30} is already in its simplest form.

Thus, the sum of 110 \frac{1}{10} and 13 \frac{1}{3} is 1330 \frac{13}{30} .

The correct answer is 1330 \frac{13}{30} , which corresponds to choice 4.

Answer

1330 \frac{13}{30}

Exercise #15

Solve the following exercise:

12+19=? \frac{1}{2}+\frac{1}{9}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we will add the fractions 12 \frac{1}{2} and 19 \frac{1}{9} by finding a common denominator.

  • First, identify the denominators: 2 and 9.
  • Find a common denominator by multiplying the denominators: 2×9=18 2 \times 9 = 18 .
  • Convert each fraction to an equivalent fraction with this common denominator:
    • Convert 12 \frac{1}{2} to have a denominator of 18 by multiplying both the numerator and denominator by 9: 1×92×9=918 \frac{1 \times 9}{2 \times 9} = \frac{9}{18} .
    • Convert 19 \frac{1}{9} to have a denominator of 18 by multiplying both the numerator and denominator by 2: 1×29×2=218 \frac{1 \times 2}{9 \times 2} = \frac{2}{18} .
  • Add the converted fractions: 918+218=1118 \frac{9}{18} + \frac{2}{18} = \frac{11}{18} .
  • The fraction 1118 \frac{11}{18} is already in its simplest form.

Thus, the sum of the fractions 12 \frac{1}{2} and 19 \frac{1}{9} is 1118 \frac{11}{18} .

Answer

1118 \frac{11}{18}

Exercise #16

Solve the following exercise:

12+25=? \frac{1}{2}+\frac{2}{5}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem of adding the fractions 12 \frac{1}{2} and 25 \frac{2}{5} , we will follow these steps:

  • Step 1: Determine a common denominator for the fractions.
  • Step 2: Convert each fraction to an equivalent fraction with this common denominator.
  • Step 3: Add the resulting fractions.

Now, let’s explore each step in detail:

Step 1: The denominators are 2 and 5. A common denominator can be found by multiplying these two numbers: 2×5=10 2 \times 5 = 10 . Therefore, 10 is our common denominator.

Step 2: Convert each fraction to have the common denominator of 10.
- For 12 \frac{1}{2} , multiply both the numerator and the denominator by 5:
12×55=510 \frac{1}{2} \times \frac{5}{5} = \frac{5}{10} .
- For 25 \frac{2}{5} , multiply both the numerator and the denominator by 2:
25×22=410 \frac{2}{5} \times \frac{2}{2} = \frac{4}{10} .

Step 3: Add the fractions 510\frac{5}{10} and 410\frac{4}{10}:
Combine the numerators while keeping the common denominator:
5+4=9 5 + 4 = 9 .
Thus, 510+410=910\frac{5}{10} + \frac{4}{10} = \frac{9}{10} .

Therefore, the sum of 12 \frac{1}{2} and 25 \frac{2}{5} is 910\frac{9}{10}.

Answer

910 \frac{9}{10}

Exercise #17

Solve the following exercise:

12+27=? \frac{1}{2}+\frac{2}{7}=\text{?}

Video Solution

Step-by-Step Solution

To solve the given problem of adding two fractions 12 \frac{1}{2} and 27 \frac{2}{7} , follow these steps:

  • Step 1: Determine the common denominator.

The denominators of the fractions are 22 and 77. Multiply these two numbers to find the common denominator: 2×7=142 \times 7 = 14.

  • Step 2: Adjust each fraction to have the common denominator.

Convert 12 \frac{1}{2} to an equivalent fraction with a denominator of 1414:
12=1×72×7=714 \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14}

Convert 27 \frac{2}{7} to an equivalent fraction with a denominator of 1414:
27=2×27×2=414 \frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}

  • Step 3: Add the adjusted fractions.

Now that both fractions have a common denominator, add them:
714+414=7+414=1114 \frac{7}{14} + \frac{4}{14} = \frac{7 + 4}{14} = \frac{11}{14}

We have successfully added the fractions and obtained the result.

Therefore, the solution to the problem is 1114 \frac{11}{14} .

Answer

1114 \frac{11}{14}

Exercise #18

Solve the following exercise:

13+24=? \frac{1}{3}+\frac{2}{4}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, let's follow these steps:

  • Step 1: Simplify the fractions if possible.
  • Step 2: Identify the common denominator.
  • Step 3: Convert each fraction to have this common denominator.
  • Step 4: Add the fractions.
  • Step 5: Simplify the result, if necessary.

Step 1: Simplify 24 \frac{2}{4} . It simplifies to 12 \frac{1}{2} .

Step 2: The denominators are now 3 and 2. Find the least common multiple of 3 and 2, which is 6.

Step 3: Convert each fraction to have the common denominator of 6:
13=1×23×2=26\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
12=1×32×3=36\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}

Step 4: Add the fractions:
26+36=2+36=56\frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6} = \frac{5}{6}

Step 5: The fraction 56\frac{5}{6} is already in its simplest form.

Therefore, the solution to the problem is 56\frac{5}{6}.

Answer

1012 \frac{10}{12}

Exercise #19

Solve the following exercise:

14+36=? \frac{1}{4}+\frac{3}{6}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem of adding 14 \frac{1}{4} and 36 \frac{3}{6} , we need to find their sum using a common denominator.

Step 1: Identify the Least Common Denominator (LCD)
The denominators of the fractions are 4 and 6. The LCM of 4 and 6, which will be the least common denominator, is 12.

Step 2: Convert each fraction to an equivalent fraction with the denominator of 12.
For 14 \frac{1}{4} : Multiply the numerator and denominator by 3 to get 1×34×3=312 \frac{1 \times 3}{4 \times 3} = \frac{3}{12} .
For 36 \frac{3}{6} : Multiply the numerator and denominator by 2 to get 3×26×2=612 \frac{3 \times 2}{6 \times 2} = \frac{6}{12} .

Step 3: Add the fractions 312+612=3+612=912 \frac{3}{12} + \frac{6}{12} = \frac{3 + 6}{12} = \frac{9}{12} .

Step 4: Simplify the resulting fraction if necessary.
In this case, 912 \frac{9}{12} can be simplified. The greatest common divisor of 9 and 12 is 3, so 912=9÷312÷3=34 \frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4} .

Therefore, the sum of 14+36 \frac{1}{4} + \frac{3}{6} is 34 \frac{3}{4} , but in the context of the provided answer choices, we are looking for 912 \frac{9}{12} initially, which does match the simplified result before reducing.

The correct answer is therefore 912 \frac{9}{12} , which corresponds to Choice 3.

Answer

912 \frac{9}{12}

Exercise #20

Solve the following exercise:

14+39=? \frac{1}{4}+\frac{3}{9}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem of adding the fractions 14 \frac{1}{4} and 39 \frac{3}{9} , we will first find a common denominator and then perform the addition:

Step 1: Finding a Common Denominator
The denominators are 4 and 9. The easiest way to find a common denominator is to multiply these two numbers. Hence, 4×9=36 4 \times 9 = 36 gives us a common denominator of 36.

Step 2: Convert Each Fraction
Convert 14 \frac{1}{4} to a fraction with denominator 36. To do this, multiply the numerator and denominator by 9 (since 4×9=364 \times 9 = 36):
14=1×94×9=936 \frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}

Next, convert 39 \frac{3}{9} to a fraction with denominator 36. Multiply the numerator and denominator by 4 (since 9×4=369 \times 4 = 36):
39=3×49×4=1236 \frac{3}{9} = \frac{3 \times 4}{9 \times 4} = \frac{12}{36}

Step 3: Add the Fractions
Now add the two fractions:
936+1236=9+1236=2136 \frac{9}{36} + \frac{12}{36} = \frac{9+12}{36} = \frac{21}{36}

Step 4: Simplify the Result (if necessary)
The fraction 2136\frac{21}{36} can be simplified by finding the greatest common divisor (GCD) of 21 and 36, which is 3. However, in the current situation with the answer choices provided, 2136\frac{21}{36} matches one of the options directly without further simplification, ensuring it meets the expected answer format.

Therefore, the sum of 14+39 \frac{1}{4} + \frac{3}{9} is 2136\frac{21}{36}, which corresponds to choice 11.

Thus, the correct answer is 2136 \frac{21}{36} .

Answer

2136 \frac{21}{36}