Multiplying and Dividing Mixed Numbers: Simplifiying fractions before multiplying

Examples with solutions for Multiplying and Dividing Mixed Numbers: Simplifiying fractions before multiplying

Exercise #1

139×224= 1\frac{3}{9}\times2\frac{2}{4}=

Video Solution

Step-by-Step Solution

To solve the problem 139×224 1\frac{3}{9} \times 2\frac{2}{4} , we will follow these steps:

  • Step 1: Convert each mixed number to an improper fraction.
  • Step 2: Multiply the improper fractions.
  • Step 3: Convert the resulting improper fraction back into a mixed number.

Let’s begin with each step in detail:

Step 1: Convert 139 1\frac{3}{9} and 224 2\frac{2}{4} to improper fractions.
- For 139 1\frac{3}{9} : Convert the fraction 39 \frac{3}{9} to its simplest form, which is 13 \frac{1}{3} . Then, the mixed number 113 1\frac{1}{3} becomes 1+13=33+13=43 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} .
- For 224 2\frac{2}{4} : The fraction 24 \frac{2}{4} simplifies to 12 \frac{1}{2} . Then, the mixed number 212 2\frac{1}{2} becomes 2+12=42+12=52 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} .

Step 2: Multiply the improper fractions:
43×52=4×53×2=206\frac{4}{3} \times \frac{5}{2} = \frac{4 \times 5}{3 \times 2} = \frac{20}{6}.

Simplify 206\frac{20}{6}:
Find the greatest common divisor (GCD) of 20 and 6, which is 2. Then 206=20÷26÷2=103\frac{20}{6} = \frac{20 \div 2}{6 \div 2} = \frac{10}{3}.

Step 3: Convert the improper fraction 103\frac{10}{3} back to a mixed number:
Divide 10 by 3 to get 3 with a remainder of 1, thus 103=313\frac{10}{3} = 3\frac{1}{3}.

Therefore, the product is 313 3\frac{1}{3} .

Answer

313 3\frac{1}{3}

Exercise #2

146×128= 1\frac{4}{6}\times1\frac{2}{8}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll convert the given mixed numbers to improper fractions, multiply them, and simplify the result. Let's proceed step by step:

  • Step 1: Convert the mixed numbers to improper fractions.
    For 1461\frac{4}{6}: Multiply the whole number by the denominator and add the numerator: 146=1×6+46=1061\frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{10}{6}.
    For 1281\frac{2}{8}: Similarly, multiply the whole number by the denominator and add the numerator: 128=1×8+28=1081\frac{2}{8} = \frac{1 \times 8 + 2}{8} = \frac{10}{8}.
  • Step 2: Multiply the improper fractions.
    106×108=10×106×8=10048 \frac{10}{6} \times \frac{10}{8} = \frac{10 \times 10}{6 \times 8} = \frac{100}{48}.
  • Step 3: Simplify the resulting fraction.
    Find the GCD of 100 and 48, which is 4. Divide both the numerator and the denominator by 4:
    10048=100÷448÷4=2512 \frac{100}{48} = \frac{100 \div 4}{48 \div 4} = \frac{25}{12}.
  • Step 4: Convert the simplified improper fraction back to a mixed number.
    Divide 25 by 12: 25 divided by 12 is 2 with a remainder of 1. So, 2512=2112\frac{25}{12} = 2\frac{1}{12}.

Therefore, the solution to the problem is 2112 2\frac{1}{12} . This matches the correct answer choice 2.

Answer

2112 2\frac{1}{12}

Exercise #3

2412×124= 2\frac{4}{12}\times1\frac{2}{4}=

Video Solution

Step-by-Step Solution

To solve the given problem, we'll follow these steps:

  • Convert the mixed numbers to improper fractions.
  • Multiply the improper fractions.
  • Simplify the result and convert back to a mixed number if necessary.

Let's work through these steps:

1. Convert each mixed number to an improper fraction:

  • For 24122\frac{4}{12}, first simplify the fraction 412\frac{4}{12} to 13\frac{1}{3}. So, 2132\frac{1}{3} becomes:
  • 2×3+13=73\frac{2 \times 3 + 1}{3} = \frac{7}{3}.

  • For 1241\frac{2}{4}, first simplify the fraction 24\frac{2}{4} to 12\frac{1}{2}. So, 1121\frac{1}{2} becomes:
  • 1×2+12=32\frac{1 \times 2 + 1}{2} = \frac{3}{2}.

2. Multiply the improper fractions:

The multiplication of 73\frac{7}{3} and 32\frac{3}{2} is:

73×32=216\frac{7}{3} \times \frac{3}{2} = \frac{21}{6}.

3. Simplify the resulting fraction 216\frac{21}{6}:

216=72\frac{21}{6} = \frac{7}{2} after dividing the numerator and denominator by 3.

4. Convert the improper fraction back to a mixed number:

72=312\frac{7}{2} = 3\frac{1}{2}.

Thus, the solution to the problem is 312\boxed{3\frac{1}{2}}.

Answer

312 3\frac{1}{2}

Exercise #4

226×1410= 2\frac{2}{6}\times1\frac{4}{10}=

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: Multiply the improper fractions.
  • Step 3: Simplify the resulting fraction, if necessary.
  • Step 4: Convert the result back to a mixed number.

Now, let's work through each step:

Step 1: Convert each mixed number to an improper fraction.

For 2262\frac{2}{6}:

226=2×6+26=12+26=146 2\frac{2}{6} = \frac{2 \times 6 + 2}{6} = \frac{12 + 2}{6} = \frac{14}{6}

Simplifying 146\frac{14}{6} by dividing the numerator and the denominator by 2, we get 146=73\frac{14}{6} = \frac{7}{3}.

For 14101\frac{4}{10}:

1410=1×10+410=10+410=1410 1\frac{4}{10} = \frac{1 \times 10 + 4}{10} = \frac{10 + 4}{10} = \frac{14}{10}

Simplifying 1410\frac{14}{10} by dividing the numerator and the denominator by 2, we get 1410=75\frac{14}{10} = \frac{7}{5}.

Step 2: Multiply the improper fractions:

73×75=7×73×5=4915 \frac{7}{3} \times \frac{7}{5} = \frac{7 \times 7}{3 \times 5} = \frac{49}{15}

Step 3: Simplify the resulting fraction, if necessary. In this case, 4915\frac{49}{15} is already in its simplest form.

Step 4: Convert the result back to a mixed number:

4915 \frac{49}{15} can be rewritten as 34153\frac{4}{15}, since 49 divided by 15 is 3 with a remainder of 4.

Therefore, the solution to the problem is 3415 3\frac{4}{15} .

Answer

3415 3\frac{4}{15}

Exercise #5

168×226= 1\frac{6}{8}\times2\frac{2}{6}=

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: Multiply the improper fractions.
  • Step 3: Simplify the resulting fraction.
  • Step 4: Convert the improper fraction back to a mixed number if necessary.

Let's begin:

Step 1: Convert to improper fractions
Convert 1681\frac{6}{8} to an improper fraction:
8×1+68=148\frac{8 \times 1 + 6}{8} = \frac{14}{8}.

Convert 2262\frac{2}{6} to an improper fraction:
6×2+26=146\frac{6 \times 2 + 2}{6} = \frac{14}{6}.

Step 2: Multiply the fractions
Multiply 148\frac{14}{8} and 146\frac{14}{6}:
148×146=19648\frac{14}{8} \times \frac{14}{6} = \frac{196}{48}.

Step 3: Simplify the fraction
Find the greatest common divisor (GCD) of 196 and 48, which is 4. Simplify 19648\frac{196}{48}:
196÷448÷4=4912\frac{196 \div 4}{48 \div 4} = \frac{49}{12}.

Step 4: Convert to a mixed number
4912\frac{49}{12} as a mixed number is 41124\frac{1}{12} since 49 divided by 12 is 4 with a remainder of 1.

Therefore, the solution to the problem is 4112 4\frac{1}{12} .

Answer

4112 4\frac{1}{12}

Exercise #6

1412×1414= 1\frac{4}{12}\times1\frac{4}{14}=

Video Solution

Step-by-Step Solution

Let's solve the problem by following these steps:

  • Step 1: Convert the mixed numbers to improper fractions.
  • Step 2: Multiply the improper fractions.
  • Step 3: Simplify the resulting fraction and convert back to a mixed number if needed.

Step 1:
Convert 14121\frac{4}{12} to an improper fraction.

Calculate the improper fraction: 1412=1×12+412=1612 1\frac{4}{12} = \frac{1 \times 12 + 4}{12} = \frac{16}{12}

Convert 14141\frac{4}{14} to an improper fraction.

Calculate the improper fraction: 1414=1×14+414=1814 1\frac{4}{14} = \frac{1 \times 14 + 4}{14} = \frac{18}{14}

Step 2:
Multiply the two improper fractions:

1612×1814=16×1812×14 \frac{16}{12} \times \frac{18}{14} = \frac{16 \times 18}{12 \times 14}

Simplify the multiplication:

  • Numerator: 16×18=28816 \times 18 = 288
  • Denominator: 12×14=16812 \times 14 = 168

The resulting fraction is 288168\frac{288}{168}.

Step 3:
Simplify 288168\frac{288}{168}.

Find the greatest common divisor (GCD) of 288 and 168, which is 24.

  • Divide numerator by GCD: 288÷24=12288 \div 24 = 12
  • Divide denominator by GCD: 168÷24=7168 \div 24 = 7

The simplified fraction is 127\frac{12}{7}.

Convert 127\frac{12}{7} back to a mixed number:

12 divided by 7 is 1 with a remainder of 5, so the mixed number is 1571\frac{5}{7}.

Thus, the solution to the problem is 157\boxed{1\frac{5}{7}}.

Answer

157 1\frac{5}{7}

Exercise #7

21020×1416= 2\frac{10}{20}\times1\frac{4}{16}=

Video Solution

Step-by-Step Solution

To solve this problem, we will convert the mixed numbers to improper fractions and multiply them.

  • Step 1: Convert Mixed Numbers to Improper Fractions

For the first mixed number 210202\frac{10}{20}:
- Convert 210202\frac{10}{20} to an improper fraction:
Initially, 10/2010/20 can be simplified to 1/21/2 (since gcd(10,20)=10\gcd(10, 20) = 10). Therefore, the mixed number 2122\frac{1}{2} is equal to:
2×2+1=4+1=52 \times 2 + 1 = 4 + 1 = 5
Thus, 2122\frac{1}{2} becomes 52\frac{5}{2}.

For the second mixed number 14161\frac{4}{16}:
- Simplify 4/164/16 to 1/41/4 (since gcd(4,16)=4\gcd(4, 16) = 4). Hence, 14161\frac{4}{16} simplifies to 1141\frac{1}{4}:
1×4+1=4+1=51 \times 4 + 1 = 4 + 1 = 5
Thus, 1141\frac{1}{4} becomes 54\frac{5}{4}.

  • Step 2: Multiply the Improper Fractions

Multiply the fractions 52×54\frac{5}{2} \times \frac{5}{4}:
5×52×4=258\frac{5 \times 5}{2 \times 4} = \frac{25}{8}

  • Step 3: Simplify and Convert to Mixed Number

Convert 258\frac{25}{8} back to a mixed number by dividing:
- Divide 25 by 8, which goes 3 times (remainder 1).
Thus, 258=318\frac{25}{8} = 3\frac{1}{8}.

Therefore, the solution to the problem is 3183\frac{1}{8}.

Answer

318 3\frac{1}{8}

Exercise #8

3515×2210= 3\frac{5}{15}\times2\frac{2}{10}=

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

  • Step 1: Convert mixed numbers to improper fractions.
  • Step 2: Simplify the fractions if feasible.
  • Step 3: Multiply the improper fractions.
  • Step 4: Convert the result back to a mixed number.

Let's work through the steps:

Step 1: Convert the mixed numbers to improper fractions.

3515=3+515=4515+515=50153\frac{5}{15} = 3 + \frac{5}{15} = \frac{45}{15} + \frac{5}{15} = \frac{50}{15}, simplifying yields 103\frac{10}{3}.

2210=2+210=2010+210=22102\frac{2}{10} = 2 + \frac{2}{10} = \frac{20}{10} + \frac{2}{10} = \frac{22}{10}, simplifying yields 115\frac{11}{5}.

Step 2: Multiply the fractions.

103×115=11015\frac{10}{3} \times \frac{11}{5} = \frac{110}{15}.

Step 3: Simplify the result of the multiplication.

11015\frac{110}{15} can be simplified by dividing the numerator and the denominator by 5, yielding 223\frac{22}{3}.

Step 4: Convert the resulting improper fraction back to a mixed number.

Dividing, 22÷3=722 \div 3 = 7 with a remainder of 1, so 223=713\frac{22}{3} = 7\frac{1}{3}.

Therefore, the solution to the problem is 7137\frac{1}{3}.

Answer

713 7\frac{1}{3}

Exercise #9

369×3420= 3\frac{6}{9}\times3\frac{4}{20}=

Video Solution

Step-by-Step Solution

To solve this problem, we will follow these steps:

  • Convert the mixed numbers to improper fractions.
  • Simplify the fractions when possible.
  • Multiply the improper fractions.
  • Convert the resulting improper fraction back to a mixed number and simplify.

Here's the step-by-step solution:

Step 1: Convert 3693 \frac{6}{9} and 34203 \frac{4}{20} to improper fractions.

For 3693 \frac{6}{9}, which can be simplified to 3233 \frac{2}{3}: 3+69=3+23=3×3+23=9+23=113 3 + \frac{6}{9} = 3 + \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} .

For 34203 \frac{4}{20}, which can be simplified to 3153 \frac{1}{5}: 3+420=3+15=3×5+15=15+15=165 3 + \frac{4}{20} = 3 + \frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5} .

Step 2: Simplify fractions if possible. The fractions 113\frac{11}{3} and 165\frac{16}{5} are already in simplest form.

Step 3: Multiply the fractions:

113×165=11×163×5=17615\frac{11}{3} \times \frac{16}{5} = \frac{11 \times 16}{3 \times 5} = \frac{176}{15}.

Step 4: Convert the resulting improper fraction to a mixed number.

Divide 176 by 15: 176÷15=11 176 \div 15 = 11 remainder 1111 . Thus, 17615=111115\frac{176}{15} = 11 \frac{11}{15}.

Therefore, the solution to the problem is 11111511 \frac{11}{15}, which matches choice 3.

Therefore, the final answer is 11111511 \frac{11}{15}.

Answer

111115 11\frac{11}{15}

Exercise #10

41012×2210= 4\frac{10}{12}\times2\frac{2}{10}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll convert the mixed numbers into improper fractions and perform the multiplication.

  • Convert 410124\frac{10}{12} into an improper fraction:
    Multiply the whole number 4 by the denominator 12, then add the numerator 10: 4×12+10=584 \times 12 + 10 = 58.
    The improper fraction is 5812\frac{58}{12}.
  • Convert 22102\frac{2}{10} into an improper fraction:
    Multiply the whole number 2 by the denominator 10, then add the numerator 2: 2×10+2=222 \times 10 + 2 = 22.
    The improper fraction is 2210\frac{22}{10}.
  • Multiply the two improper fractions:
    5812×2210=58×2212×10=1276120\frac{58}{12} \times \frac{22}{10} = \frac{58 \times 22}{12 \times 10} = \frac{1276}{120}.
  • Simplify the result:
    First, find the greatest common divisor (GCD) of 1276 and 120, which is 4.
    Divide both numerator and denominator by 4 to simplify:
    1276÷4120÷4=31930\frac{1276 \div 4}{120 \div 4} = \frac{319}{30}.
  • Convert 31930\frac{319}{30} into a mixed number:
    Divide 319 by 30: the quotient is 10 and the remainder is 19.
    Thus, 31930=101930\frac{319}{30} = 10\frac{19}{30}.

Therefore, the solution to the problem is 101930 10\frac{19}{30} , which corresponds to choice 3.

Answer

101930 10\frac{19}{30}

Exercise #11

128×1714= 1\frac{2}{8}\times1\frac{7}{14}=

Video Solution

Step-by-Step Solution

To solve the problem 128×1714 1\frac{2}{8} \times 1\frac{7}{14} , follow these steps:

  • Step 1: Convert the Mixed Numbers to Improper Fractions
    For 128 1\frac{2}{8} , the improper fraction is calculated as:
    128=1+28=88+28=108 1\frac{2}{8} = 1 + \frac{2}{8} = \frac{8}{8} + \frac{2}{8} = \frac{10}{8}
  • For 1714 1\frac{7}{14} , simplify 714=12 \frac{7}{14} = \frac{1}{2} , hence: 1714=1+12=22+12=32 1\frac{7}{14} = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}
  • Step 2: Multiply the Improper Fractions
    Multiply the fractions 108\frac{10}{8} and 32\frac{3}{2}:
    108×32=10×38×2=3016\frac{10}{8} \times \frac{3}{2} = \frac{10 \times 3}{8 \times 2} = \frac{30}{16}
  • Step 3: Simplify the Resulting Fraction
    Simplify 3016\frac{30}{16} by finding the greatest common divisor (GCD) of 30 and 16, which is 2:
    3016=30÷216÷2=158\frac{30}{16} = \frac{30 \div 2}{16 \div 2} = \frac{15}{8}
  • Step 4: Convert Back to a Mixed Number
    Convert 158\frac{15}{8} to a mixed number:
    158=178\frac{15}{8} = 1\frac{7}{8}

Therefore, the solution to the problem is 178 1\frac{7}{8} .

Answer

178 1\frac{7}{8}

Exercise #12

346×224= 3\frac{4}{6}\times2\frac{2}{4}=

Video Solution

Step-by-Step Solution

To solve the problem of multiplying these mixed fractions, we will proceed through the following steps:

  • Step 1: Convert each mixed number into an improper fraction.
    3463\frac{4}{6} must first be simplified to 3233\frac{2}{3}.
    To convert 3233\frac{2}{3} to an improper fraction: 3×3+2=113 \times 3 + 2 = 11, so we have 113\frac{11}{3}.
    For 2242\frac{2}{4}, simplify 24\frac{2}{4} to 12\frac{1}{2} making the fraction 2122\frac{1}{2}.
    Convert 2122\frac{1}{2} to an improper fraction: 2×2+1=52 \times 2 + 1 = 5, so we have 52\frac{5}{2}.
  • Step 2: Multiply the two improper fractions:
    The calculation is 113×52=11×53×2=556\frac{11}{3} \times \frac{5}{2} = \frac{11 \times 5}{3 \times 2} = \frac{55}{6}.
  • Step 3: Simplify 556\frac{55}{6} by converting it back to a mixed number.
    Divide 55 by 6 which gives 9 as the quotient and 1 as the remainder. Therefore, 556=916\frac{55}{6} = 9\frac{1}{6}.

Therefore, the result of the multiplication is 9169\frac{1}{6}.

Answer

916 9\frac{1}{6}