Examples with solutions for Addition and Subtraction of Mixed Numbers: One of the denominators is the common denominator

Exercise #1

328134= 3\frac{2}{8}-1\frac{3}{4}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Convert 1341\frac{3}{4} to a fraction with a denominator of 8.
  • Step 2: Subtract the whole numbers and fractions separately.
  • Step 3: Simplify the resulting fraction and combine it with the whole number part.

Now, let's work through each step:

Step 1: Convert 1341\frac{3}{4} to a fraction with a denominator of 8.
Since 34=68 \frac{3}{4} = \frac{6}{8} (by multiplying numerator and denominator by 2), we have:
134=1+68=88+68=1481\frac{3}{4} = 1 + \frac{6}{8} = \frac{8}{8} + \frac{6}{8} = \frac{14}{8}.

Step 2: Subtract the whole numbers and fractions separately.
Original problem: 328134=2681483\frac{2}{8} - 1\frac{3}{4} = \frac{26}{8} - \frac{14}{8}.

Subtract the fractions:
268148=128\frac{26}{8} - \frac{14}{8} = \frac{12}{8}.

Step 3: Simplify the resulting fraction and combine it with the whole number part.
128=32\frac{12}{8} = \frac{3}{2}.

Therefore, after subtracting and simplifying, we find that:

Therefore, the solution to the problem is 112 1\frac{1}{2} .

Answer

112 1\frac{1}{2}

Exercise #2

243112= 2\frac{4}{3}-1\frac{1}{2}=

Video Solution

Step-by-Step Solution

To solve the problem 2431122\frac{4}{3} - 1\frac{1}{2}, let's follow these steps:

  • Step 1: Convert the mixed numbers into improper fractions.
  • Step 2: Find a common denominator for the fractions.
  • Step 3: Perform the subtraction of the fractions.
  • Step 4: Simplify the resulting fraction if possible, and convert it back to a mixed number if needed.

Now let's work through these steps:

Step 1: Convert the mixed numbers into improper fractions.

243=2×3+43=6+43=1032\frac{4}{3} = \frac{2 \times 3 + 4}{3} = \frac{6 + 4}{3} = \frac{10}{3}

112=1×2+12=2+12=321\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}

Step 2: Identify a common denominator. The denominators are 3 and 2; the least common denominator is 6.

Convert 103\frac{10}{3} to have a denominator of 6:

103×22=206\frac{10}{3} \times \frac{2}{2} = \frac{20}{6}

Convert 32\frac{3}{2} to have a denominator of 6:

32×33=96\frac{3}{2} \times \frac{3}{3} = \frac{9}{6}

Step 3: Perform the subtraction:

20696=2096=116\frac{20}{6} - \frac{9}{6} = \frac{20 - 9}{6} = \frac{11}{6}

Step 4: Convert 116\frac{11}{6} back to a mixed number:

116=156\frac{11}{6} = 1\frac{5}{6}

Therefore, the solution to the problem is 1561\frac{5}{6}.

Answer

156 1\frac{5}{6}

Exercise #3

456+1112= 4\frac{5}{6}+1\frac{1}{12}=

Video Solution

Step-by-Step Solution

To solve the addition of the mixed numbers 456 4\frac{5}{6} and 1112 1\frac{1}{12} , follow these steps:

  • Step 1: Add the whole numbers.
    The whole numbers are 4 and 1, so we add these together:
    4+1=5 4 + 1 = 5 .
  • Step 2: Add the fractional parts.
    First, find a common denominator for the fractions 56\frac{5}{6} and 112\frac{1}{12}. The least common denominator of 6 and 12 is 12.
    Convert 56\frac{5}{6} to twelfths: 56=5×26×2=1012\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}.
    Now add the fractions: 1012+112=1112\frac{10}{12} + \frac{1}{12} = \frac{11}{12}.
  • Step 3: Combine the results.
    Add the whole number and the sum of the fractions:
    5+1112=511125 + \frac{11}{12} = 5\frac{11}{12}.

Thus, the sum of 456+1112 4\frac{5}{6} + 1\frac{1}{12} is 51112 5\frac{11}{12} .

Answer

51112 5\frac{11}{12}

Exercise #4

114+212= 1\frac{1}{4}+2\frac{1}{2}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Convert the mixed numbers to improper fractions.
  • Step 2: Find a common denominator and perform the addition.
  • Step 3: Convert the resulting improper fraction back to a mixed number.

Now, let's work through each step:
Step 1: Convert each mixed number to an improper fraction:

  • 114 1\frac{1}{4} becomes 54 \frac{5}{4} because 1×4+1=5 1 \times 4 + 1 = 5 .
  • 212 2\frac{1}{2} becomes 52 \frac{5}{2} because 2×2+1=5 2 \times 2 + 1 = 5 .

Step 2: Find a common denominator and add the fractions:
The least common denominator of 4 and 2 is 4.
Convert 52 \frac{5}{2} to 104 \frac{10}{4} by multiplying the numerator and the denominator by 2.
Now, add the fractions: 54+104=154 \frac{5}{4} + \frac{10}{4} = \frac{15}{4} .

Step 3: Convert the improper fraction back to a mixed number:
Divide 15 by 4. The whole number is 3, and the remainder is 3.
So, 154=334 \frac{15}{4} = 3\frac{3}{4} .

Therefore, the solution to the problem is 334 3\frac{3}{4} .

Answer

334 3\frac{3}{4}

Exercise #5

123+126= 1\frac{2}{3}+1\frac{2}{6}=

Video Solution

Step-by-Step Solution

To solve this problem, we will proceed with the following steps:

  • Step 1: Identify the whole and fractional parts of each mixed number.

  • Step 2: Convert the fractions to have a common denominator.

  • Step 3: Add the fractional parts.

  • Step 4: Add the whole numbers.

  • Step 5: Combine the results to obtain the total.

Now, let's go through the solution:

Step 1: Separate the mixed numbers into whole and fractional parts:

  • 123=1+23 1\frac{2}{3} = 1 + \frac{2}{3}

  • 126=1+26 1\frac{2}{6} = 1 + \frac{2}{6}

Step 2: Convert the fractions 23\frac{2}{3} and 26\frac{2}{6} to have a common denominator. The least common multiple of 3 and 6 is 6.

Convert 23\frac{2}{3}:

23=2×23×2=46\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}

Step 3: Add the fractional parts:

46+26=4+26=66=1\frac{4}{6} + \frac{2}{6} = \frac{4+2}{6} = \frac{6}{6} = 1

Step 4: Add the whole number parts:

1+1=21 + 1 = 2

Step 5: Combine the whole and fractional results:

Adding the whole sum and fractional sum gives: 2+1=32 + 1 = 3

Therefore, the sum of 123+126 1\frac{2}{3} + 1\frac{2}{6} is 3 3 .

Answer

3 3

Exercise #6

31321415= 3\frac{1}{3}-2\frac{14}{15}=

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

  • Step 1: Convert each mixed number to an improper fraction.
  • Step 2: Find a common denominator for the fractions.
  • Step 3: Subtract the fractions.
  • Step 4: Simplify the result if needed.

Let's begin with Step 1:
Convert 3133\frac{1}{3} to an improper fraction:
313=3×3+13=1033\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}
Convert 214152\frac{14}{15} to an improper fraction:
21415=2×15+1415=44152\frac{14}{15} = \frac{2 \times 15 + 14}{15} = \frac{44}{15}

Step 2: Find a common denominator for 103\frac{10}{3} and 4415\frac{44}{15}.
The least common multiple of 3 and 15 is 15, so we'll use 15 as the common denominator.

Step 3: Convert 103\frac{10}{3} to have the denominator of 15:
103=10×53×5=5015\frac{10}{3} = \frac{10 \times 5}{3 \times 5} = \frac{50}{15}

Now, perform the subtraction:
50154415=504415=615\frac{50}{15} - \frac{44}{15} = \frac{50 - 44}{15} = \frac{6}{15}

Step 4: Simplify 615\frac{6}{15}.
The greatest common divisor of 6 and 15 is 3, so divide the numerator and denominator by 3:
615=6÷315÷3=25\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}

Thus, the final answer is 25\frac{2}{5}.

Answer

25 \frac{2}{5}

Exercise #7

619223= 6\frac{1}{9}-2\frac{2}{3}=

Video Solution

Step-by-Step Solution

To solve this problem, we will follow a structured approach to subtract the given mixed numbers:

  • Step 1: Convert mixed numbers to improper fractions.

  • Step 2: Find a common denominator for the fractions.

  • Step 3: Subtract the improper fractions.

  • Step 4: Convert the result back to a mixed number if necessary.

Let's begin with each step in detail:
Step 1: Convert each mixed number to an improper fraction.
- 6196\frac{1}{9} becomes 559\frac{55}{9} because 6×9+1=556 \times 9 + 1 = 55.
- 2232\frac{2}{3} becomes 83\frac{8}{3} because 2×3+2=82 \times 3 + 2 = 8.

Step 2: Convert 83\frac{8}{3} to have a common denominator with 559\frac{55}{9}.
The least common denominator (LCD) is 9.
Multiply the numerator and the denominator of 83\frac{8}{3} by 3 to get 249\frac{24}{9}.

Step 3: Subtract the improper fractions:
559249=319\frac{55}{9} - \frac{24}{9} = \frac{31}{9}.

Step 4: Convert the improper fraction back to a mixed number.
319\frac{31}{9} simplifies to 3493\frac{4}{9}, because 31÷931 \div 9 equals 3 with a remainder of 4.

Therefore, the solution to the problem is 349 3\frac{4}{9} .

Answer

349 3\frac{4}{9}

Exercise #8

335+1115= 3\frac{3}{5}+1\frac{1}{15}=

Video Solution

Step-by-Step Solution

To solve the problem 335+11153\frac{3}{5} + 1\frac{1}{15}, we'll follow these steps:

  • Step 1: Convert each mixed number to an improper fraction.
    - For 3353\frac{3}{5}: Multiply the whole number 33 by the denominator 55 and add the numerator 33: (3×5)+3=15+3=18(3 \times 5) + 3 = 15 + 3 = 18, so we have 185\frac{18}{5}.
    - For 11151\frac{1}{15}: Multiply the whole number 11 by the denominator 1515 and add the numerator 11: (1×15)+1=15+1=16(1 \times 15) + 1 = 15 + 1 = 16, so we have 1615\frac{16}{15}.
  • Step 2: Find the least common denominator (LCD).
    - The denominators are 55 and 1515, the LCD is 1515.
  • Step 3: Convert fractions to have the same denominator.
    - 185=18×35×3=5415\frac{18}{5} = \frac{18 \times 3}{5 \times 3} = \frac{54}{15}
    - 1615\frac{16}{15} is already expressed with the denominator 1515.
  • Step 4: Add the fractions.
    - 5415+1615=54+1615=7015\frac{54}{15} + \frac{16}{15} = \frac{54 + 16}{15} = \frac{70}{15}
  • Step 5: Simplify the fraction.
    - Simplify 7015 \frac{70}{15} . The greatest common divisor of 7070 and 1515 is 55. So, 7015=70÷515÷5=143\frac{70}{15} = \frac{70 \div 5}{15 \div 5} = \frac{14}{3}.
  • Step 6: Convert the improper fraction back to a mixed number.
    - 143=423 \frac{14}{3} = 4\frac{2}{3} (since 14÷3=414 \div 3 = 4 remainder 22)

Therefore, the sum of 335+11153\frac{3}{5} + 1\frac{1}{15} is 423 \mathbf{4\frac{2}{3}} .

Answer

423 4\frac{2}{3}

Exercise #9

3516+658= 3\frac{5}{16}+6\frac{5}{8}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Convert fractions to a common denominator.
  • Step 2: Add the fractions.
  • Step 3: Add the whole numbers and the resulting fraction.
  • Step 4: Simplify the result.

Now, let's work through each step:

Step 1: Convert 58 \frac{5}{8} to a fraction with denominator 16.
Currently, 58=5×28×2=1016 \frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16} .

Step 2: Add the fractions 516 \frac{5}{16} and 1016 \frac{10}{16} .
516+1016=1516 \frac{5}{16} + \frac{10}{16} = \frac{15}{16} .

Step 3: Add the whole numbers of the mixed numbers.
The whole numbers are 3 and 6, giving 3+6=9 3 + 6 = 9 .

Step 4: Combine the results of steps 2 and 3 into a mixed number.
9+1516=91516 9 + \frac{15}{16} = 9\frac{15}{16} .

Therefore, the solution to the problem is 91516 9\frac{15}{16} , which corresponds to choice 3 in the provided options.

Answer

91516 9\frac{15}{16}

Exercise #10

12149+237= 12\frac{1}{49}+2\frac{3}{7}=

Video Solution

Step-by-Step Solution

To solve the problem of adding 12149+237 12\frac{1}{49} + 2\frac{3}{7} , we'll follow step-by-step procedures:

  • Step 1: Identify the whole numbers and fractions:
    Whole numbers: 12 and 2.
    Fractions: 149\frac{1}{49} and 37\frac{3}{7}.
  • Step 2: Convert 37\frac{3}{7} to a fraction with a denominator of 49:
    Multiply both the numerator and denominator by 7: 3×77×7=2149\frac{3 \times 7}{7 \times 7} = \frac{21}{49}.
  • Step 3: Add the fractions:
    149+2149=1+2149=2249\frac{1}{49} + \frac{21}{49} = \frac{1 + 21}{49} = \frac{22}{49}.
  • Step 4: Add the whole numbers:
    12 + 2 = 14.
  • Step 5: Combine the whole number result and the fraction result:
    The result is 142249 14\frac{22}{49} .

Therefore, the solution to the problem 12149+237 12\frac{1}{49} + 2\frac{3}{7} is 142249 14\frac{22}{49} , which matches choice 4.

Answer

142249 14\frac{22}{49}

Exercise #11

756+623+13= ? 7\frac{5}{6}+6\frac{2}{3}+\frac{1}{3}=\text{ ?}

Video Solution

Step-by-Step Solution

Note that the right-hand side of the addition exercise between the fractions gives a result of a whole number, so we'll start with that:

623+13=7 6\frac{2}{3}+\frac{1}{3}=7

Giving us:

756+7=1456 7\frac{5}{6}+7=14\frac{5}{6}

Answer

1456 14\frac{5}{6}