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

Exercise #1

Solve the following exercise:

1315=? \frac{1}{3}-\frac{1}{5}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem 1315 \frac{1}{3} - \frac{1}{5} , we follow these steps:

First, we need to find a common denominator for the fractions 13\frac{1}{3} and 15\frac{1}{5}. The denominators are 3 and 5, and their least common multiple (LCM) is 15.

We will convert each fraction to an equivalent fraction with the denominator 15:

  • To convert 13\frac{1}{3} to a fraction with denominator 15, multiply both the numerator and the denominator by 5: 13=1×53×5=515 \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
  • To convert 15\frac{1}{5} to a fraction with denominator 15, multiply both the numerator and the denominator by 3: 15=1×35×3=315 \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}

Now that both fractions have the same denominator, we can subtract the numerators:

515315=5315=215 \frac{5}{15} - \frac{3}{15} = \frac{5 - 3}{15} = \frac{2}{15}

Therefore, the solution to the problem is 215\frac{2}{15}.

Answer

215 \frac{2}{15}

Exercise #2

Solve the following exercise:

3716=? \frac{3}{7}-\frac{1}{6}=\text{?}

Video Solution

Step-by-Step Solution

To solve the subtraction problem 3716 \frac{3}{7} - \frac{1}{6} , we need to follow these steps:

  • Step 1: Identify the least common multiple (LCM) of the denominators. For 7 and 6, the LCM is 42.
  • Step 2: Convert each fraction to an equivalent fraction with a denominator of 42.
    • For 37 \frac{3}{7} , multiply the numerator and the denominator by 6 to get 1842 \frac{18}{42} .
    • For 16 \frac{1}{6} , multiply the numerator and the denominator by 7 to get 742 \frac{7}{42} .
  • Step 3: Subtract the two fractions: 1842742=1142 \frac{18}{42} - \frac{7}{42} = \frac{11}{42} .
  • Step 4: Check if the fraction 1142 \frac{11}{42} can be simplified. Since 11 and 42 have no common factors besides 1, the fraction is already in its simplest form.

Therefore, the solution to the problem is 1142 \frac{11}{42} .

Among the provided choices, the correct answer is: 1142\frac{11}{42}.

Answer

1142 \frac{11}{42}

Exercise #3

Solve the following exercise:

2413=? \frac{2}{4}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Find the common denominator for the fractions 24\frac{2}{4} and 13\frac{1}{3}.
  • Step 2: Convert each fraction to have the common denominator.
  • Step 3: Perform the subtraction and simplify if necessary.

Now, let's work through these steps:

Step 1: The denominators are 44 and 33. The common denominator is the product 4×3=124 \times 3 = 12.

Step 2: Convert each fraction:
24=2×34×3=612\frac{2}{4} = \frac{2 \times 3}{4 \times 3} = \frac{6}{12}
13=1×43×4=412\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}

Step 3: Subtract the fractions with a common denominator:
612412=6412=212\frac{6}{12} - \frac{4}{12} = \frac{6 - 4}{12} = \frac{2}{12}

Finally, simplify 212\frac{2}{12}. The greatest common divisor of 2 and 12 is 2, so:
212=2÷212÷2=16\frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6}

Therefore, the solution to the problem is 16\frac{1}{6}.

Answer

16 \frac{1}{6}

Exercise #4

Solve the following exercise:

3512=? \frac{3}{5}-\frac{1}{2}=\text{?}

Video Solution

Step-by-Step Solution

To solve the subtraction of fractions 3512 \frac{3}{5} - \frac{1}{2} , we will follow these steps:

  • Step 1: Find the least common multiple (LCM) of the denominators 5 and 2. The LCM of 5 and 2 is 10.
  • Step 2: Convert each fraction to have a denominator of 10.
  • Step 3: Subtract the converted fractions.
  • Step 4: Simplify the result if necessary.

Now, let's work through each step in detail:

Step 1: The LCM of 5 and 2 is 10, since 10 is the smallest number that both 5 and 2 divide into evenly.

Step 2: Convert each fraction to have a denominator of 10.

For 35\frac{3}{5}:
Multiply numerator and denominator by 2 to get 3×25×2=610\frac{3 \times 2}{5 \times 2} = \frac{6}{10}.

For 12\frac{1}{2}:
Multiply numerator and denominator by 5 to get 1×52×5=510\frac{1 \times 5}{2 \times 5} = \frac{5}{10}.

Step 3: Subtract the fractions:

610510=6510=110\frac{6}{10} - \frac{5}{10} = \frac{6 - 5}{10} = \frac{1}{10}.

Step 4: There is no further simplification needed for 110\frac{1}{10} as it is already in its simplest form.

Therefore, the solution to the problem is 110\frac{1}{10}.

The correct answer, choice (4), is 110\frac{1}{10}.

Answer

110 \frac{1}{10}

Exercise #5

Solve the following exercise:

3514=? \frac{3}{5}-\frac{1}{4}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem of subtracting 14 \frac{1}{4} from 35 \frac{3}{5} , we need a common denominator.

First, find the least common denominator (LCD) of 5 and 4, which is 20. This is done by multiplying the denominators: 5×4=20 5 \times 4 = 20 .

Next, convert each fraction to an equivalent fraction with the denominator of 20:

  • For 35 \frac{3}{5} : Multiply both numerator and denominator by 4 to get 3×45×4=1220 \frac{3 \times 4}{5 \times 4} = \frac{12}{20} .
  • For 14 \frac{1}{4} : Multiply both numerator and denominator by 5 to get 1×54×5=520 \frac{1 \times 5}{4 \times 5} = \frac{5}{20} .

Now perform the subtraction with these equivalent fractions:

1220520=12520=720 \frac{12}{20} - \frac{5}{20} = \frac{12 - 5}{20} = \frac{7}{20}

The resulting fraction, 720 \frac{7}{20} , is already in its simplest form.

Therefore, the solution to the subtraction 3514 \frac{3}{5} - \frac{1}{4} is 720 \frac{7}{20} .

Checking against the multiple-choice answers, the correct choice is the first one: 720 \frac{7}{20} .

Answer

720 \frac{7}{20}

Exercise #6

Solve the following exercise:

1227=? \frac{1}{2}-\frac{2}{7}=\text{?}

Video Solution

Step-by-Step Solution

To solve the given problem 1227 \frac{1}{2} - \frac{2}{7} , we need to follow these steps:

  • Step 1: Find a common denominator for the fractions. The denominators are 2 and 7. The common denominator can be found by multiplying these two numbers: 2×7=14 2 \times 7 = 14 .
  • Step 2: Convert the fractions to equivalent fractions with the common denominator. For 12 \frac{1}{2} , multiply the numerator and the denominator by 7: 12=1×72×7=714 \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14} . For 27 \frac{2}{7} , multiply the numerator and the denominator by 2: 27=2×27×2=414 \frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14} .
  • Step 3: Subtract the fractions. With a common denominator, subtract the numerators: 714414=7414=314 \frac{7}{14} - \frac{4}{14} = \frac{7 - 4}{14} = \frac{3}{14} .
  • Step 4: Simplify the resulting fraction if needed. The fraction 314\frac{3}{14} is already in its simplest form.

Thus, the solution to the problem is 314 \frac{3}{14} .

Answer

314 \frac{3}{14}

Exercise #7

Solve the following exercise:

3813=? \frac{3}{8}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve the problem of subtracting 13 \frac{1}{3} from 38 \frac{3}{8} , follow these steps:

Step 1: Identify the denominators of the fractions, which are 8 and 3, respectively. The least common denominator (LCD) is the product of these two denominators, as they have no common factors. Thus, the LCD is 8×3=24 8 \times 3 = 24 .

Step 2: Convert each fraction to an equivalent form with the common denominator 24.

  • Convert 38\frac{3}{8} to an equivalent fraction with a denominator of 24 by multiplying both the numerator and the denominator by 248=3\frac{24}{8} = 3:
    38×33=924\frac{3}{8} \times \frac{3}{3} = \frac{9}{24}.
  • Convert 13\frac{1}{3} to an equivalent fraction with a denominator of 24 by multiplying both the numerator and the denominator by 243=8\frac{24}{3} = 8:
    13×88=824\frac{1}{3} \times \frac{8}{8} = \frac{8}{24}.

Step 3: Subtract the numerators of these equivalent fractions, maintaining the common denominator:

924824=9824=124\frac{9}{24} - \frac{8}{24} = \frac{9 - 8}{24} = \frac{1}{24}.

Therefore, the solution to 3813\frac{3}{8} - \frac{1}{3} is 124\frac{1}{24}.

Answer

124 \frac{1}{24}

Exercise #8

Solve the following exercise:

2526=? \frac{2}{5}-\frac{2}{6}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll walk through these steps:

  • Step 1: Determine the least common denominator (LCD) for the denominators 5 and 6.
  • Step 2: Convert each fraction to an equivalent fraction with the LCD.
  • Step 3: Subtract the numerators of the equivalent fractions.
  • Step 4: Simplify the resulting fraction if possible.

Now, let's work through each step:

Step 1: Determine the least common denominator (LCD).
The denominators are 5 and 6. Since there is no common factor, the LCD is 5×6=30 5 \times 6 = 30 .

Step 2: Convert each fraction to an equivalent fraction with the LCD.
For 25 \frac{2}{5} , multiply numerator and denominator by 6 to get 2×65×6=1230 \frac{2 \times 6}{5 \times 6} = \frac{12}{30} .
For 26 \frac{2}{6} , multiply numerator and denominator by 5 to get 2×56×5=1030 \frac{2 \times 5}{6 \times 5} = \frac{10}{30} .

Step 3: Subtract the fractions with the same denominator.
12301030=121030=230 \frac{12}{30} - \frac{10}{30} = \frac{12 - 10}{30} = \frac{2}{30} .

Step 4: Simplify the resulting fraction.
230 \frac{2}{30} can be simplified by dividing numerator and denominator by their greatest common divisor, which is 2.
Thus, 230=115 \frac{2}{30} = \frac{1}{15} .

Therefore, the solution to the problem is 115 \frac{1}{15} .

Answer

115 \frac{1}{15}

Exercise #9

Solve the following exercise:

3713=? \frac{3}{7}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the fractions and denominators involved.
  • Step 2: Find a common denominator for the two fractions.
  • Step 3: Convert each fraction to an equivalent fraction with the common denominator.
  • Step 4: Subtract the numerators and simplify the result.

Let's perform each of these steps:

Step 1: We have the fractions 37\frac{3}{7} and 13\frac{1}{3}.

Step 2: Find a common denominator. The denominators are 7 and 3, so the common denominator will be 7×3=217 \times 3 = 21.

Step 3: Convert each fraction:

  • For 37\frac{3}{7}, multiply both numerator and denominator by 3 to get 3×37×3=921\frac{3 \times 3}{7 \times 3} = \frac{9}{21}.
  • For 13\frac{1}{3}, multiply both numerator and denominator by 7 to get 1×73×7=721\frac{1 \times 7}{3 \times 7} = \frac{7}{21}.

Step 4: Subtract the numerators:

921721=9721=221\frac{9}{21} - \frac{7}{21} = \frac{9 - 7}{21} = \frac{2}{21}.

Simplify if necessary: Here, 221\frac{2}{21} is already in its simplest form.

Therefore, the solution to the problem is 221 \frac{2}{21} .

Answer

221 \frac{2}{21}

Exercise #10

Solve the following exercise:

3439=? \frac{3}{4}-\frac{3}{9}=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Find a common denominator for the fractions 34 \frac{3}{4} and 39 \frac{3}{9} .
  • Step 2: Convert each fraction to have this common denominator.
  • Step 3: Subtract the numerators while keeping the common denominator.
  • Step 4: Simplify the resultant fraction if possible.

Now, let's work through each step:

Step 1: The denominators of the fractions are 4 and 9. The least common multiple of 4 and 9 is 36. Therefore, 36 will be our common denominator.

Step 2: Convert 34 \frac{3}{4} and 39 \frac{3}{9} to have a denominator of 36:

34×99=2736 \frac{3}{4} \times \frac{9}{9} = \frac{27}{36} and 39×44=1236 \frac{3}{9} \times \frac{4}{4} = \frac{12}{36}

Step 3: Now, subtract the fractions:

27361236=1536 \frac{27}{36} - \frac{12}{36} = \frac{15}{36}

Step 4: Simplify 1536 \frac{15}{36} :

Both 15 and 36 can be divided by their greatest common divisor, which is 3. Dividing both the numerator and denominator by 3, we get:

15÷336÷3=512 \frac{15 \div 3}{36 \div 3} = \frac{5}{12}

Therefore, the solution to the problem is 512 \frac{5}{12} .

Answer

512 \frac{5}{12}

Exercise #11

Solve the following exercise:

4513=? \frac{4}{5}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve the subtraction of these two fractions, we'll follow these steps:

  • Step 1: Find a common denominator for the fractions 45 \frac{4}{5} and 13 \frac{1}{3} . The denominators are 5 and 3, and multiplying them gives us a common denominator of 15.
  • Step 2: Convert each fraction to have this common denominator:
    • Convert 45 \frac{4}{5} to an equivalent fraction with a denominator of 15. To do this, multiply both the numerator and the denominator by 3: 45=4×35×3=1215\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}.
    • Convert 13 \frac{1}{3} to an equivalent fraction with a denominator of 15. To do this, multiply both the numerator and the denominator by 5: 13=1×53×5=515\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}.
  • Step 3: Perform the subtraction of the numerators, now having the same denominator: 1215515=12515=715\frac{12}{15} - \frac{5}{15} = \frac{12 - 5}{15} = \frac{7}{15}.
  • Step 4: Simplify the result, if possible. In this case, 715\frac{7}{15} is already in its simplest form as 7 and 15 have no common factors other than 1.

Therefore, the solution to the problem is 715\frac{7}{15}.

Answer

715 \frac{7}{15}

Exercise #12

Solve the following exercise:

3513=? \frac{3}{5}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve the subtraction of fractions 3513 \frac{3}{5} - \frac{1}{3} , follow these steps:

  • Step 1: Find the Least Common Denominator (LCD)
    The denominators are 5 and 3. The least common multiple of 5 and 3 is 15. Thus, the common denominator will be 15.
  • Step 2: Convert fractions to have the same denominator
    For 35 \frac{3}{5} , multiply both the numerator and the denominator by 3 to get:
    35=3×35×3=915\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}.
    For 13 \frac{1}{3} , multiply both the numerator and the denominator by 5 to get:
    13=1×53×5=515\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}.
  • Step 3: Subtract the numerators
    Now subtract the equivalent fractions:
    915515=9515=415\frac{9}{15} - \frac{5}{15} = \frac{9 - 5}{15} = \frac{4}{15}.
  • Step 4: Simplify the fraction
    The fraction 415\frac{4}{15} is already in its simplest form.

Thus, the solution to the problem is 415\frac{4}{15}.

Answer

415 \frac{4}{15}

Exercise #13

Solve the following exercise:

3436=? \frac{3}{4}-\frac{3}{6}=\text{?}

Video Solution

Step-by-Step Solution

To solve the subtraction of these two fractions, we follow these steps:

  • Step 1: Find the least common multiple (LCM) of the denominators 4 and 6. The LCM of 4 and 6 is 12.
  • Step 2: Convert each fraction to have this common denominator:
    • For 34\frac{3}{4}, multiply both the numerator and denominator by 3 to get 912\frac{9}{12}.
    • For 36\frac{3}{6}, multiply both the numerator and denominator by 2 to get 612\frac{6}{12}.
  • Step 3: Perform the subtraction: 912612=312\frac{9}{12} - \frac{6}{12} = \frac{3}{12}.
  • Step 4: Simplify the fraction 312\frac{3}{12}. The greatest common divisor (GCD) of 3 and 12 is 3. Divide both numerator and denominator by 3 to get 14\frac{1}{4}.

Therefore, the solution to the problem is 14 \frac{1}{4} .

Answer

14 \frac{1}{4}

Exercise #14

Solve the following exercise:

1219=? \frac{1}{2}-\frac{1}{9}=\text{?}

Video Solution

Step-by-Step Solution

To solve 1219\frac{1}{2} - \frac{1}{9}, follow these steps:

Step 1: Find the least common multiple (LCM) of the denominators 2 and 9.
The multiples of 2 are 2,4,6,8,10,12,14,16,18,2, 4, 6, 8, 10, 12, 14, 16, 18, \ldots
The multiples of 9 are 9,18,27,9, 18, 27, \ldots
The smallest common multiple is 18. Thus, the LCM of 2 and 9 is 18.

Step 2: Convert each fraction to an equivalent fraction with the common denominator 18.
For 12\frac{1}{2}, the equivalent fraction with 18 as the denominator is calculated by finding the factor needed:
12=1×92×9=918 \frac{1}{2} = \frac{1 \times 9}{2 \times 9} = \frac{9}{18} .
For 19\frac{1}{9}, the equivalent fraction with 18 as the denominator is:
19=1×29×2=218 \frac{1}{9} = \frac{1 \times 2}{9 \times 2} = \frac{2}{18} .

Step 3: Perform the subtraction of these equivalent fractions.
918218=9218=718 \frac{9}{18} - \frac{2}{18} = \frac{9 - 2}{18} = \frac{7}{18} .

Therefore, the solution to the problem is 718\boxed{\frac{7}{18}}.

Answer

718 \frac{7}{18}

Exercise #15

Solve the following exercise:

61013=? \frac{6}{10}-\frac{1}{3}=\text{?}

Video Solution

Step-by-Step Solution

To solve 61013 \frac{6}{10} - \frac{1}{3} , we need to perform the following steps:

  • Step 1: Identify a common denominator. The denominators 10 and 3 have a least common multiple of 30. We will use 30 as the common denominator.
  • Step 2: Convert the fractions to have the common denominator of 30.
    610\frac{6}{10} needs to be converted by multiplying both the numerator and the denominator by 3: 6×310×3=1830 \frac{6 \times 3}{10 \times 3} = \frac{18}{30} 13\frac{1}{3} needs to be converted by multiplying both the numerator and the denominator by 10: 1×103×10=1030 \frac{1 \times 10}{3 \times 10} = \frac{10}{30}
  • Step 3: Subtract the new fractions: 18301030=181030=830 \frac{18}{30} - \frac{10}{30} = \frac{18 - 10}{30} = \frac{8}{30}
  • Step 4: Simplify the fraction 830\frac{8}{30}.
    Divide both the numerator and the denominator by their greatest common divisor, which is 2: 8÷230÷2=415 \frac{8 \div 2}{30 \div 2} = \frac{4}{15}

Therefore, the result of the subtraction is 415 \frac{4}{15} .

Answer

415 \frac{4}{15}