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

Exercise #1

Fill in the missing sign:

3419 \frac{3}{4}☐\frac{1}{9}

Video Solution

Step-by-Step Solution

To determine the correct inequality sign to compare 34\frac{3}{4} and 19\frac{1}{9}, we'll use the technique of cross-multiplication. This method involves multiplying the numerator of each fraction by the denominator of the other fraction. By calculating and comparing these products, we can determine the relative size of the fractions.

Let's perform the cross-multiplication:

  • Multiply the numerator of the first fraction by the denominator of the second fraction:
 3×9=27\ 3 \times 9 = 27
  • Multiply the denominator of the first fraction by the numerator of the second fraction:
 4×1=4\ 4 \times 1 = 4

Now we compare these two results. Since 2727 is greater than 44, we conclude that:

 34>19\ \frac{3}{4} > \frac{1}{9}

Therefore, the correct inequality sign to fill in the blank is >>.

Answer

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Exercise #2

Fill in the missing sign:

13310 \frac{1}{3}☐\frac{3}{10}

Video Solution

Step-by-Step Solution

To solve this problem, we'll compare the fractions 13 \frac{1}{3} and 310 \frac{3}{10} by finding a common denominator.

Let's follow these steps:

  • Step 1: Identify the denominators of the given fractions, which are 3 and 10.
  • Step 2: Calculate the least common denominator (LCD) of 3 and 10. Since 3 and 10 are co-prime, their product, 30, is the LCD.
  • Step 3: Adjust the fractions to have a common denominator of 30:
    • Convert 13 \frac{1}{3} to a fraction with denominator 30 by multiplying the numerator and the denominator by 10:
    • 13=1×103×10=1030\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}
    • Convert 310 \frac{3}{10} to a fraction with denominator 30 by multiplying the numerator and the denominator by 3:
    • 310=3×310×3=930\frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30}
  • Step 4: Compare the numerators of the fractions: 10 (from 1030 \frac{10}{30} ) and 9 (from 930 \frac{9}{30} ).
  • Since 10 is greater than 9, 1030 \frac{10}{30} is greater than 930 \frac{9}{30} .
  • Thus, 13 \frac{1}{3} is greater than 310 \frac{3}{10} .

Therefore, the correct comparison sign is > > .

Answer

>

Exercise #3

Fill in the missing sign:

3426 \frac{3}{4}☐\frac{2}{6}

Video Solution

Step-by-Step Solution

To solve this problem, we will follow these steps:

  • Step 1: Identify the fractions we need to compare, which are 34 \frac{3}{4} and 26 \frac{2}{6} .
  • Step 2: Simplify 26 \frac{2}{6} to its simplest form.
  • Step 3: Find a common denominator for the two fractions.
  • Step 4: Convert each fraction to have the common denominator.
  • Step 5: Compare the resulting numerators to determine the relationship.

Now, let's work through each step:

Step 1: The fractions we have are 34 \frac{3}{4} and 26 \frac{2}{6} .

Step 2: Simplify 26 \frac{2}{6} . The greatest common factor of 2 and 6 is 2, so 26=13 \frac{2}{6} = \frac{1}{3} .

Step 3: Find a common denominator for 34 \frac{3}{4} and 13 \frac{1}{3} . The least common multiple of 4 and 3 is 12.

Step 4: Convert each fraction to have the common denominator:

34=3×34×3=912 \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

13=1×43×4=412 \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}

Step 5: Compare the numerators of the converted fractions:

Now, compare 912 \frac{9}{12} and 412 \frac{4}{12} .

Since 9>4 9 > 4 , it follows that 912>412 \frac{9}{12} > \frac{4}{12} .

Therefore, 34>13 \frac{3}{4} > \frac{1}{3} , and hence 34>26 \frac{3}{4} > \frac{2}{6} .

The correct comparison sign is > > .

Answer

>

Exercise #4

Fill in the missing sign:

1534 \frac{1}{5}☐\frac{3}{4}

Video Solution

Step-by-Step Solution

To compare the fractions 15 \frac{1}{5} and 34 \frac{3}{4} , we'll use the following steps:

  • Step 1: Find a common denominator for the fractions.
  • Step 2: Convert each fraction to an equivalent fraction with the common denominator.
  • Step 3: Compare the numerators of the equivalent fractions.

Step 1: The denominators of the two fractions are 5 and 4. To find a common denominator, we multiply these together, getting 5×4=20 5 \times 4 = 20 . So, our common denominator is 20.

Step 2: Convert each fraction to have the denominator of 20.

  • 15\frac{1}{5}: Multiply the numerator and denominator by 4 to get 1×45×4=420\frac{1 \times 4}{5 \times 4} = \frac{4}{20}.
  • 34\frac{3}{4}: Multiply the numerator and denominator by 5 to get 3×54×5=1520\frac{3 \times 5}{4 \times 5} = \frac{15}{20}.

Step 3: Now compare the numerators of the equivalent fractions:

420\frac{4}{20} compared to 1520\frac{15}{20} (both having the denominator of 20):

4<154 < 15 implies 420<1520\frac{4}{20} < \frac{15}{20}.

Therefore, 15<34\frac{1}{5} < \frac{3}{4}.

Thus, the correct mathematical sign to fill in is < < .

Therefore, the missing sign in the expression 1534\frac{1}{5} ☐ \frac{3}{4} is <<.

Answer

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Exercise #5

Fill in the missing sign:

2364 \frac{2}{3}☐\frac{6}{4}

Video Solution

Step-by-Step Solution

To solve this problem, we will use cross-multiplication to compare the two fractions 23\frac{2}{3} and 64\frac{6}{4}.

  • Step 1: Calculate the cross products. Multiply the numerator of the first fraction by the denominator of the second fraction, and the denominator of the first fraction by the numerator of the second fraction.
  • Step 2: Compare the two products to determine which fraction is larger.

Now, let's work through the steps:

Step 1: Cross-multiply the two fractions:
First Product=2×4=8\text{First Product} = 2 \times 4 = 8
Second Product=3×6=18\text{Second Product} = 3 \times 6 = 18

Step 2: Compare the resulting products:
Since 8<188 < 18, it follows that 23<64\frac{2}{3} < \frac{6}{4}.

Therefore, the solution to the problem is <<.

Answer

<

Exercise #6

Fill in the missing sign:

21513 \frac{2}{15}☐\frac{1}{3}

Video Solution

Step-by-Step Solution

To solve this problem, we'll convert both fractions to have a common denominator and compare:

  • Step 1: Identify the denominators (15(15 and 3)3) of the fractions 215\frac{2}{15} and 13\frac{1}{3}.
  • Step 2: Find the common denominator. The least common multiple (LCM) of 1515 and 33 is 1515.
  • Step 3: Convert each fraction to have the common denominator 1515.
    • For 215\frac{2}{15}, the fraction is already over 1515, so it stays 215\frac{2}{15}.
    • For 13\frac{1}{3}, multiply both the numerator and the denominator by 55 to get an equivalent fraction: 1×53×5=515\frac{1 \times 5}{3 \times 5} = \frac{5}{15}.
  • Step 4: Compare the equivalent fractions 215\frac{2}{15} and 515\frac{5}{15}.
  • Step 5: Since 2<52 \lt 5, it follows that 215<515\frac{2}{15} \lt \frac{5}{15}.

Therefore, the correct sign to fill in the missing slot is <\lt.

Consequently, the completed expression is 215<13\frac{2}{15} \lt \frac{1}{3}.

Therefore, the solution to the problem is < \lt .

Answer

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Exercise #7

Fill in the missing sign:

3716 \frac{3}{7}☐\frac{1}{6}

Video Solution

Step-by-Step Solution

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

  • Step 1: Determine a common denominator for the fractions.
  • Step 2: Convert each fraction to have the common denominator.
  • Step 3: Compare the numerators to find the correct inequality.

Now, let's work through each step:
Step 1: Determine a common denominator for 37\frac{3}{7} and 16\frac{1}{6}. The common denominator is the product of the denominators 7×6=427 \times 6 = 42.
Step 2: Convert each fraction to have this common denominator:
- Convert 37\frac{3}{7}: Multiply the numerator and denominator by 6: 3×67×6=1842\frac{3 \times 6}{7 \times 6} = \frac{18}{42}.
- Convert 16\frac{1}{6}: Multiply the numerator and denominator by 7: 1×76×7=742\frac{1 \times 7}{6 \times 7} = \frac{7}{42}.
Step 3: Compare the numerators: 1818 and 77.
Since 18>718 > 7, we have 1842>742\frac{18}{42} > \frac{7}{42}. Thus, 37>16\frac{3}{7} > \frac{1}{6}.

Therefore, the correct inequality sign is > > .

Answer

>

Exercise #8

Fill in the missing sign:

29314 \frac{2}{9}☐\frac{3}{14}

Video Solution

Step-by-Step Solution

To solve this problem, we'll compare the fractions 29 \frac{2}{9} and 314 \frac{3}{14} by finding a common denominator:

  • Step 1: Find the common denominator. The denominators are 9 9 and 14 14 . The least common multiple of 9 9 and 14 14 is 126 126 .
  • Step 2: Convert each fraction to an equivalent fraction with a denominator of 126 126 .
    For 29 \frac{2}{9} : 29=2×149×14=28126\frac{2}{9} = \frac{2 \times 14}{9 \times 14} = \frac{28}{126}.
    For 314 \frac{3}{14} : 314=3×914×9=27126\frac{3}{14} = \frac{3 \times 9}{14 \times 9} = \frac{27}{126}.
  • Step 3: Compare the numerators of the equivalent fractions. 28>27 28 > 27 , so 28126>27126 \frac{28}{126} > \frac{27}{126} .

Therefore, the fraction 29 \frac{2}{9} is greater than 314 \frac{3}{14} , and the correct inequality sign is > > .

Hence, the answer is 29>314\frac{2}{9} > \frac{3}{14} .

Answer

>

Exercise #9

Fill in the missing sign:

3819 \frac{3}{8}☐\frac{1}{9}

Video Solution

Step-by-Step Solution

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

  • Step 1: Calculate the least common denominator of the fractions
  • Step 2: Convert both fractions to have this common denominator
  • Step 3: Compare the numerators of the fractions with the common denominator
  • Step 4: Select the appropriate inequality sign

Now, let's work through each step:

Step 1: The denominators of the given fractions are 8 and 9. The least common multiple (LCM) of these numbers is 72. Therefore, the least common denominator (LCD) is 72.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 72.

  • For 38\frac{3}{8}: Multiply the numerator and the denominator by 9 (since 728=9 \frac{72}{8} = 9 ). This gives 3×98×9=2772\frac{3 \times 9}{8 \times 9} = \frac{27}{72}.
  • For 19\frac{1}{9}: Multiply the numerator and the denominator by 8 (since 729=8 \frac{72}{9} = 8 ). This gives 1×89×8=872\frac{1 \times 8}{9 \times 8} = \frac{8}{72}.

Step 3: Now compare the numerators of 2772\frac{27}{72} and 872\frac{8}{72}.

Since 2727 is greater than 88, it follows that 2772\frac{27}{72} is greater than 872\frac{8}{72}.

Step 4: Therefore, the correct inequality sign to fill the blank is >>.

Thus, we conclude that 38>19\frac{3}{8} > \frac{1}{9}.

The correct answer is >\gt.

Answer

>

Exercise #10

Fill in the missing sign:

3718 \frac{3}{7}☐\frac{1}{8}

Video Solution

Step-by-Step Solution

To solve this problem, we will compare the fractions 37\frac{3}{7} and 18\frac{1}{8} by converting them to have a common denominator.

Step 1: Find the least common multiple (LCM) of the denominators 7 and 8. Since 7 and 8 are coprime (have no common factors other than 1), the LCM is simply the product of the two numbers:

LCM(7,8)=7×8=56 \text{LCM}(7, 8) = 7 \times 8 = 56

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 56.

  • Convert 37\frac{3}{7}:

  • 37=3×87×8=2456 \frac{3}{7} = \frac{3 \times 8}{7 \times 8} = \frac{24}{56}

  • Convert 18\frac{1}{8}:

  • 18=1×78×7=756 \frac{1}{8} = \frac{1 \times 7}{8 \times 7} = \frac{7}{56}

Step 3: Compare the new numerators:

2456and756 \frac{24}{56} \quad \text{and} \quad \frac{7}{56}

Since 24 > 7, we conclude that \frac{24}{56} > \frac{7}{56}.

Therefore, the original inequality we are solving is:

\frac{3}{7} > \frac{1}{8}

Thus, the correct sign to fill in the blank is \bm{>}.

Answer

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