Examples with solutions for Addition and Subtraction of Mixed Numbers: The common denominator is smaller than the product of the denominators

Exercise #1

638+234= 6\frac{3}{8}+2\frac{3}{4}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Convert each mixed number to an improper fraction.
  • Step 2: Find a common denominator for the fractions.
  • Step 3: Add the fractions and simplify the result.
  • Step 4: Convert the result back to a mixed number if necessary.

Now, let's work through each step:

Step 1: Convert mixed numbers to improper fractions.

The first mixed number is 6386\frac{3}{8}. To convert this:

  • The whole number part is 6, and the fractional part is 38\frac{3}{8}.
  • Multiply the whole number by the denominator: 6×8=486 \times 8 = 48.
  • Add the numerator: 48+3=5148 + 3 = 51.
  • The improper fraction is 518\frac{51}{8}.

The second mixed number is 2342\frac{3}{4}. To convert this:

  • The whole number part is 2, and the fractional part is 34\frac{3}{4}.
  • Multiply the whole number by the denominator: 2×4=82 \times 4 = 8.
  • Add the numerator: 8+3=118 + 3 = 11.
  • The improper fraction is 114\frac{11}{4}.

Step 2: Find a common denominator for 518\frac{51}{8} and 114\frac{11}{4}.

The denominators are 8 and 4. The least common denominator is 8.

Step 3: Add the fractions.

Convert 114\frac{11}{4} to the equivalent fraction with a denominator of 8:

  • Multiply both the numerator and the denominator by 2: 11×24×2=228\frac{11 \times 2}{4 \times 2} = \frac{22}{8}.

Now add: 518+228=51+228=738\frac{51}{8} + \frac{22}{8} = \frac{51 + 22}{8} = \frac{73}{8}.

Step 4: Convert 738\frac{73}{8} back to a mixed number.

  • Divide 73 by 8: 73÷8=973 \div 8 = 9 remainder 1.
  • The quotient is 9, and the remainder is 1.
  • The mixed number is 9189\frac{1}{8}.

Therefore, the solution to the problem is 9189\frac{1}{8}.

Answer

918 9\frac{1}{8}

Exercise #2

724+314= 7\frac{2}{4}+3\frac{1}{4}=

Video Solution

Step-by-Step Solution

To solve the problem of adding 724+3147\frac{2}{4} + 3\frac{1}{4}, we follow these steps:

  • Step 1: Add the whole numbers from both mixed numbers: 7+3=107 + 3 = 10.
  • Step 2: Add the fractional parts. Since the denominators are the same, we simply add the numerators: 24+14=34\frac{2}{4} + \frac{1}{4} = \frac{3}{4}.
  • Step 3: Combine the results: The sum of the whole numbers is 1010 and the sum of the fractions is 34\frac{3}{4}. Therefore, the total is 103410\frac{3}{4}.

The final result of the mixed number addition 724+3147\frac{2}{4} + 3\frac{1}{4} is 103410\frac{3}{4}.

Answer

1034 10\frac{3}{4}

Exercise #3

216+223= 2\frac{1}{6}+2\frac{2}{3}=

Video Solution

Step-by-Step Solution

To find the sum of 216+223 2\frac{1}{6} + 2\frac{2}{3} , we will follow these steps:

  • Step 1: Separate the mixed numbers into their whole and fractional parts:
    • Whole parts: 22 and 22.
    • Fractional parts: 16 \frac{1}{6} and 23 \frac{2}{3} .
  • Step 2: Add the whole parts:
    • 2+2=42 + 2 = 4.
  • Step 3: Convert fractional parts using a common denominator:
    • The fractions are 16 \frac{1}{6} and 23 \frac{2}{3} .
    • The least common denominator of 66 and 33 is 66.
    • 23\frac{2}{3} is converted to 46\frac{4}{6} by multiplying both numerator and denominator by 22.
  • Step 4: Add the fractions:
    • 16+46=56 \frac{1}{6} + \frac{4}{6} = \frac{5}{6} .
  • Step 5: Combine results:
    • Combine the whole and fractional parts: 4+56=4564 + \frac{5}{6} = 4\frac{5}{6}.

Therefore, the sum of the mixed numbers 216 2\frac{1}{6} and 223 2\frac{2}{3} is 456 4\frac{5}{6} .

Answer

456 4\frac{5}{6}

Exercise #4

312+834= 3\frac{1}{2}+8\frac{3}{4}=

Video Solution

Step-by-Step Solution

To solve the problem of adding 312+834 3\frac{1}{2} + 8\frac{3}{4} , we'll go through these steps:

  • Step 1: Separate and add the whole numbers.
  • Step 2: Add the fractional parts by finding a common denominator.
  • Step 3: Combine the results and simplify if necessary.

Let's apply these steps in detail:

Step 1: Add the whole numbers:
3+8=11 3 + 8 = 11 .

Step 2: Add the fractions 12 \frac{1}{2} and 34 \frac{3}{4} :
The denominators are 2 and 4. The least common denominator is 4.

Convert 12 \frac{1}{2} to an equivalent fraction with a denominator of 4:
12=1×22×2=24 \frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} .

Now, add the fractions:
24+34=2+34=54 \frac{2}{4} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4} .

54 \frac{5}{4} is an improper fraction, which can be converted to a mixed number:
54=114 \frac{5}{4} = 1\frac{1}{4} .

Step 3: Combine the whole numbers and the mixed fraction:
11+114=1214 11 + 1\frac{1}{4} = 12\frac{1}{4} .

Therefore, the solution to the problem is 1214 12\frac{1}{4} .

Answer

1214 12\frac{1}{4}

Exercise #5

612134= 6\frac{1}{2}-1\frac{3}{4}=

Video Solution

Step-by-Step Solution

Let's work through the problem:

Step 1: Convert to improper fractions.
The mixed number 6126\frac{1}{2} is converted to an improper fraction as follows:

  • Multiply the whole number 6 by the denominator 2: 6×2=126 \times 2 = 12.

  • Add the numerator 1: 12+1=1312 + 1 = 13.

  • The improper fraction is 132\frac{13}{2}.

The mixed number 1341\frac{3}{4} is converted to an improper fraction as follows:

  • Multiply the whole number 1 by the denominator 4: 1×4=41 \times 4 = 4.

  • Add the numerator 3: 4+3=74 + 3 = 7.

  • The improper fraction is 74\frac{7}{4}.

Step 2: Find a common denominator.
The denominators are 2 and 4. The least common denominator is 4.

Step 3: Adjust fractions to have the common denominator.
132\frac{13}{2} needs to be adjusted to have a denominator of 4:

  • Multiply both the numerator and the denominator by 2: 13×22×2=264\frac{13 \times 2}{2 \times 2} = \frac{26}{4}.

Step 4: Perform the subtraction.
Subtract 74\frac{7}{4} from 264\frac{26}{4}:

  • 26474=2674=194\frac{26}{4} - \frac{7}{4} = \frac{26 - 7}{4} = \frac{19}{4}.

Step 5: Convert the improper fraction back to a mixed number.
Divide the numerator by the denominator:

  • 19÷4=419 \div 4 = 4 with a remainder of 3.

  • The mixed number is 4344\frac{3}{4}.

Therefore, the solution to the problem is 4344\frac{3}{4}.

Answer

434 4\frac{3}{4}

Exercise #6

3715215= 3\frac{7}{15}-2\frac{1}{5}=

Video Solution

Step-by-Step Solution

To solve this problem, we will subtract the mixed numbers 37153\frac{7}{15} and 2152\frac{1}{5} by considering their whole number parts and fractions separately.

Step 1: Subtract the Whole Numbers
The whole number part of the first mixed number is 3, and for the second mixed number, it is 2. Subtracting these gives:

32=1 3 - 2 = 1

Step 2: Subtract the Fractional Parts
The fractional part of the first number is 715\frac{7}{15}, and for the second, it is 15\frac{1}{5}. We need a common denominator to subtract these fractions. The least common denominator for 15 and 5 is 15. Convert 15\frac{1}{5} to a fraction with denominator 15:

15=1×35×3=315 \frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}

Now, subtract the fractions:

715315=7315=415 \frac{7}{15} - \frac{3}{15} = \frac{7 - 3}{15} = \frac{4}{15}

Step 3: Combine the Whole Number and Fractional Parts
The whole number part is 1, and the fractional part is 415\frac{4}{15}. Combining these gives:

1415 1\frac{4}{15}

Thus, the result of the subtraction 37152153\frac{7}{15} - 2\frac{1}{5} is:

14151\frac{4}{15}

Therefore, the solution to the problem is 14151\frac{4}{15}.

Answer

1415 1\frac{4}{15}

Exercise #7

6514217= 6\frac{5}{14}-2\frac{1}{7}=

Video Solution

Step-by-Step Solution

To solve the problem 65142176\frac{5}{14} - 2\frac{1}{7}, we'll follow these steps:

  • Step 1: Identify a common denominator for the fractions. The denominators are 14 and 7, and the least common denominator is 14.
  • Step 2: Convert 2172\frac{1}{7} to a fraction with a denominator of 14 to match 65146\frac{5}{14}.
  • Step 3: Subtract the fractions.
  • Step 4: Subtract the whole numbers.

Now, let's work through each step:

Step 1: The fractions are 514 \frac{5}{14} and 17 \frac{1}{7} . The common denominator is 14.

Step 2: Convert 2172\frac{1}{7}: Start by converting 17\frac{1}{7} to a denominator of 14:
17 \frac{1}{7} is equivalent to 214 \frac{2}{14} (since 1×2=21 \times 2 = 2 and 7×2=147 \times 2 = 14).
So, 217=22142\frac{1}{7} = 2\frac{2}{14}.

Step 3: Subtract the fractions:
Subtract 214 \frac{2}{14} from 514 \frac{5}{14} :
514214=314 \frac{5}{14} - \frac{2}{14} = \frac{3}{14} .

Step 4: Subtract the whole numbers:
62=46 - 2 = 4.

Therefore, the final result is:
4314 4\frac{3}{14} .

The correct answer to the problem 65142176\frac{5}{14} - 2\frac{1}{7} is 4314 4\frac{3}{14} .

Answer

4314 4\frac{3}{14}

Exercise #8

279+113= 2\frac{7}{9}+1\frac{1}{3}=

Video Solution

Step-by-Step Solution

Let's solve the problem by converting the mixed numbers into improper fractions, finding a common denominator, and adding the fractions.

Step 1: Convert Mixed Numbers to Improper Fractions
- 2792\frac{7}{9} becomes 2×9+79=18+79=259\frac{2 \times 9 + 7}{9} = \frac{18 + 7}{9} = \frac{25}{9}.
- 1131\frac{1}{3} becomes 1×3+13=3+13=43\frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}.

Step 2: Find a Common Denominator
- The denominators are 9 and 3. The least common denominator is 9.

Step 3: Convert Fractions to a Common Denominator
- 259\frac{25}{9} is already with the denominator 9.
- 43=4×33×3=129\frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9}.

Step 4: Add the Fractions
259+129=25+129=379\frac{25}{9} + \frac{12}{9} = \frac{25 + 12}{9} = \frac{37}{9}.

Step 5: Convert the Improper Fraction Back to a Mixed Number
- 379 \frac{37}{9} is 4194 \frac{1}{9} because 37 divided by 9 is 4 with a remainder of 1.

Therefore, the solution to the problem 279+113 2\frac{7}{9}+1\frac{1}{3} is 419 4\frac{1}{9} .

Answer

419 4\frac{1}{9}

Exercise #9

13310+212= 13\frac{3}{10}+2\frac{1}{2}=

Video Solution

Step-by-Step Solution

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

  • Step 1: Identify the integer and fractional parts of each mixed number.
  • Step 2: Find a common denominator for the fractional parts.
  • Step 3: Add the fractional parts.
  • Step 4: Add the integer parts.
  • Step 5: Combine the results from Steps 3 and 4.

Now, let's work through each step:

Step 1: Identify the integer and fractional parts:
1331013\frac{3}{10} has an integer part of 13 and a fractional part of 310\frac{3}{10}.
2122\frac{1}{2} has an integer part of 2 and a fractional part of 12\frac{1}{2}.

Step 2: Find a common denominator for 310\frac{3}{10} and 12\frac{1}{2}.
The denominators are 10 and 2, with the common denominator being 10. Convert 12\frac{1}{2} to an equivalent fraction with this common denominator: 12=510\frac{1}{2} = \frac{5}{10}.

Step 3: Add the fractional parts:
310+510=810\frac{3}{10} + \frac{5}{10} = \frac{8}{10}.

Step 4: Add the integer parts:
13+2=1513 + 2 = 15.

Step 5: Combine the integer sum and fractional sum:
The result is 1581015\frac{8}{10}.

Therefore, the solution to the problem is 1581015\frac{8}{10}.

Answer

15810 15\frac{8}{10}

Exercise #10

312+126= 3\frac{1}{2}+1\frac{2}{6}=

Video Solution

Step-by-Step Solution

Let's solve the problem by adding the mixed numbers step-by-step:

  • Step 1: Add the whole numbers. The whole numbers are 33 and 11, so 3+1=43 + 1 = 4.
  • Step 2: Add the fractional parts. The fractions are 12\frac{1}{2} and 26\frac{2}{6}.
  • Step 3: Find a common denominator for the fractions. The denominators are 22 and 66. The least common denominator (LCD) is 66.
  • Step 4: Convert 12\frac{1}{2} to a fraction with the common denominator: 12=36\frac{1}{2} = \frac{3}{6}.
  • Step 5: The fractions 36\frac{3}{6} and 26\frac{2}{6} can now be added: 36+26=56\frac{3}{6} + \frac{2}{6} = \frac{5}{6}.
  • Step 6: Combine the results from Steps 1 and 5: 4+56=4564 + \frac{5}{6} = 4\frac{5}{6}.

Therefore, the result of adding 312+1263\frac{1}{2} + 1\frac{2}{6} is 456 4\frac{5}{6} .

Answer

456 4\frac{5}{6}