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Adding Mixed Numbers with Unlike Denominators

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Adding Mixed Numbers with Unlike Denominators
CCSS.MATH.CONTENT.5.NF.A.1

Basics on the topic Adding Mixed Numbers with Unlike Denominators

Adding Mixed Numbers and Unlike Denominators

When we dive into the world of fractions, we encounter different forms that numbers can take. Among these are mixed numbers and fractions with unlike denominators. These concepts are crucial for understanding advanced arithmetic operations and problem-solving in mathematics.

Adding Mixed Numbers and Unlike Denominators – Definition

Mixed numbers are those that consist of a whole number and a fraction combined, such as $2 \frac{3}{4}$. They are especially common in everyday life – think of recipes, measurements, and timekeeping.

Converting Mixed Numbers to Improper Fractions

To perform calculations with mixed numbers, such as addition or subtraction, we often need to convert them into improper fractions, where the numerator (the top number) is larger than the denominator (the bottom number).

How do you convert the mixed number $4 \frac{1}{3}$ into an improper fraction?
If you have the mixed number $7 \frac{2}{5}$, what is the equivalent improper fraction?
Convert the mixed number $3 \frac{3}{8}$ into an improper fraction.

Adding Mixed Numbers and Unlike Denominators – Explanation

When dealing with fractions, unlike denominators refer to two or more fractions that have different numbers at the bottom. Unlike denominators must be made the same, or common, before we can add or subtract the fractions.

Step Description Details
1 Convert each mixed number into an improper fraction. $1 \frac{2}{3}$ becomes $\frac{5}{3}$
$2 \frac{1}{4}$ becomes $\frac{9}{4}$
2 Find the least common denominator (LCD) for the fractions. LCD for $\frac{5}{3}$ and $\frac{9}{4}$
3 Add the improper fractions and simplify if necessary. Add $\frac{5}{3}$ and $\frac{9}{4}$, simplify the result

Adding Mixed Numbers with Unlike Denominators – Exercises

Calculate the sum of the mixed numbers $2 \frac{1}{6}$ and $3 \frac{1}{3}$.
Add the mixed numbers $4 \frac{2}{5}$ and $1 \frac{3}{4}$.
What is the sum of $7 \frac{1}{8}$ and $5 \frac{2}{5}$?
Sum the mixed numbers $5 \frac{3}{7}$ and $2 \frac{1}{2}$.
Find the total of $6 \frac{1}{3}$ and $3 \frac{3}{8}$.

Mixed Numbers and Unlike Denominators – Summary

Key Learnings from this Text:

  • Mixed numbers combine a whole number with a fractional part.

  • Unlike denominators in fractions need to be converted to a common denominator before performing addition or subtraction.

  • To work with mixed numbers in calculations, first convert them to improper fractions.

Practice with more examples and explore our resources to further develop your understanding of mixed numbers and unlike denominators.

Mixed Numbers and Unlike Denominators – Frequently Asked Questions

What is a mixed number?
How do you convert a mixed number to an improper fraction?
Why do you need to find a common denominator to add fractions with unlike denominators?
Can you add mixed numbers without converting to improper fractions?
How do you simplify an improper fraction back into a mixed number?
What's the first step in adding mixed numbers with unlike denominators?
Is it necessary to always find the least common denominator when adding fractions with unlike denominators?
How can mixed numbers be used in real life?
What do you do if adding the fractions' numerators results in an improper fraction?
Are there any shortcuts to adding mixed numbers?

Transcript Adding Mixed Numbers with Unlike Denominators

Today is the perfect day! The pouring rain gives us the best excuse to stay indoors. And what's even more exciting? The long-awaited new season of demise has just been released! Time to binge-watch before the spoilers come in and ruin it! We can track how much time we spend watching our favorite show by adding mixed numbers with unlike denominators. Time can be converted into mixed numbers that show fractional amounts that represent parts of a whole, as in hours, or in minutes that represent a portion of an hour. We can add fractional amounts of time to determine the time spent on an activity, like binge watching a series. Let's say it took us one and three-fourths hours to watch episode one, and one and half hours for episode two. We set up the problem like this. To add these mixed numbers, we can to convert them to improper fractions. Starting with this fraction, we multiply the denominator, four by one, and add three to get seven-fourths. In this fraction, we multiply the two by one and add one to get three halves. Now we need to find the least common denominator of four and two, which is four. Seven-fourths would remain the same, and we would multiply three halves by two. The renamed fraction is sixth-fourths. To add these fractions, we bring down the denominator, and add the numerators to get thirteen-fourths. We need to convert this improper fraction back to a mixed number. We do this by dividing the denominator into the numerator. Four divides into thirteen three times, so this becomes our whole number. Four times three is twelve, and thirteen take away twelve is one. One becomes the numerator, and the denominator, four, remains the same. The episodes were watched in three and one-fourth hours. Back to the show for two and three-fourths hours, and after a snack break, one and two-thirds hours more. What is the total number of hours spent on episodes three and four? Let's walk through this one together. You can pause the video at any point if you need extended time. First, convert the mixed numbers to improper fractions. What are the renamed fractions? Eleven-fourths and five-thirds. Now find the least common denominator of four and three. Twelve. What are the renamed fractions with the new numerators? Thirty-three twelfths, twenty-twelfths. What do you get when you add the fractions? Fifty-three twelfths. And now convert it back to a mixed number by dividing the denominator into the numerator. Six and three-twelfths. There is one more step to this answer. Can you see what it is? Three-twelfths can be simplified because three and twelve can be divided by three. What is the final answer for the time spent? Six and one-fourth hours.

Adding Mixed Numbers with Unlike Denominators exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Adding Mixed Numbers with Unlike Denominators .
  • Can you convert these times to mixed numbers?

    Hints

    The minutes represent part of an hour. There are 60 minutes in one hour so start by putting the number of minutes over 60.

    For example, 30 minutes would be $\frac{30}{60}$.

    Simplify the fractions by finding a common factor.

    1 hour 30 minutes is equal to $1 \frac{30}{60}$.

    If you divide both 30 and 60 by 30, what is the simplified fraction?

    Solution
    • 1 hour 30 minutes = $1 \frac{30}{60}$ = $1 \frac{1}{2}$
    ${}$
    • 2 hours 20 minutes = $2 \frac{20}{60}$ = $2 \frac{1}{3}$
    ${}$
    • 2 hours 35 minutes = $2 \frac{35}{60}$ = $2 \frac{7}{12}$
    ${}$
    • 1 hour 12 minutes = $1 \frac{12}{60}$ = $1 \frac{1}{5}$
  • How long has Luis been watching TV for?

    Hints

    The addition problem is:

    $1 \frac{1}{12} + 1 \frac{3}{4}$

    We can convert these mixed numbers to improper fractions to help with addition.

    $1 \frac{1}{12}$ $+$ $1 \frac{3}{4}$

    is equal to

    $\frac{13}{12}$ $+$ $\frac{7}{4}$

    Now find a common denominator.

    $\frac{13}{12}$ $+$ $\frac{7}{4}$

    is equal to

    $\frac{13}{12}$ $+$ $\frac{21}{12}$

    $\frac{13}{12}$ $+$ $\frac{21}{12}$ $=$ $\frac{34}{12}$

    Convert $\frac{34}{12}$ to a mixed number and simplify.

    Solution

    The correct answer is $2 \frac{5}{6}$ hours.

    $1 \frac{1}{12} + 1 \frac{3}{4}$

    is equal to

    $\frac{13}{12} + \frac{7}{4}$

    is equal to

    $\frac{13}{12} + \frac{21}{12} = \frac{34}{12} = 2 \frac{10}{12} = 2 \frac{5}{6}$

  • Which two programmes has Luis watched?

    Hints

    Start by converting $2 \frac{5}{6}$ to an improper fraction.

    $2 \frac{5}{6} = \frac{17}{6}$ or $\frac{34}{12}$

    Now convert all of the programme times to improper fractions.

    Which two would add together to equal $\frac{34}{12}$?

    • Football Today = $\frac{20}{12}$
    ${}$
    • Laughathon = $\frac{26}{12}$
    ${}$
    • The Pinnacle = $\frac{13}{12}$
    ${}$
    • The Ultimate Game Show = $\frac{14}{12}$
    Solution

    The correct answers are Football Today and The Ultimate Game Show.

    Luis has been watching TV for $2 \frac{5}{6}$ hours which is equal to $\frac{17}{6}$ or $\frac{34}{12}$.

    If we convert all of the programme time lengths to improper fractions with a denominator of $12$ we get:

    • Football Today = $\frac{20}{12}$
    ${}$
    • Laughathon = $\frac{26}{12}$
    ${}$
    • The Pinnacle = $\frac{13}{12}$
    ${}$
    • The Ultimate Game Show = $\frac{14}{12}$
    ${}$

    We can see that if we add Football Today = $\frac{20}{12}$ + The Ultimate Game Show = $\frac{14}{12}$ we get $\frac{34}{12}$.

  • Which programmes could Luis watch if he had exactly $2 \frac{5}{12}$ hours?

    Hints

    What is $2 \frac{5}{12}$ as in improper fraction?

    Convert all of the programme lengths to improper fractions.

    Make sure all of the improper fractions have the same denominator.

    Which two programmes add together to equal $2 \frac{5}{12}$ as in improper fraction?

    There are two programmes to highlight.

    Solution

    The correct answers are The Home Show and The Seven Seas.

    The total amount of time Luis had was $2 \frac{5}{12}$ or $\frac{29}{12}$.

    The programme lengths are:

    • Cricket : $2 \frac{1}{12}$ hours = $\frac{25}{12}$
    • The Home Show : $1 \frac{1}{6}$ hours = $\frac{7}{6}$ or $\frac{14}{12}$
    • The Seven Seas : $1 \frac{1}{4}$ hours = $\frac{5}{4}$ or $\frac{15}{12}$
    • History and More : $1 \frac{7}{12}$ hours = $\frac{19}{12}$
    • News Special : $1 \frac{2}{3}$ hours = $\frac{5}{3}$ or $\frac{20}{12}$
    ${}$

    The only two programmes that add up to $\frac{29}{12}$ are The Home Show $\frac{14}{12}$ and The Seven Seas $\frac{15}{12}$.

  • Complete the addition problem.

    Hints

    First convert the mixed numbers to improper fractions.

    • $1 \times 6 = 6 + 5 = 11$ therefore $1 \frac{5}{6} = \frac{?}{6}$
    • $3 \times 3 = 9 + 1 = 10$ therefore $3 \frac{1}{3} = \frac{?}{3}$

    $3 \frac{1}{3} = \frac{10}{3} = \frac{?}{6}$

    We multiplied the denominator $3$ by two to get $6$ so multiply the numerator $10$ by two as well.

    $\frac{11}{6} + \frac{20}{6} = \frac{?}{6}$

    Convert this answer to a mixed number.

    Solution

    Here we can see the completed addition problem.

    We first needed to convert the mixed numbers to improper fractions:

    • $1 \times 6 = 6$
    • $6 + 5 = 11$ therefore $1 \frac{5}{6} = \frac{11}{6}$
    • $3 \times 3 = 9$
    • $9 + 1 = 10$ therefore $3 \frac{1}{3} = \frac{10}{3}$
    We then need these fractions to have the same denominator so we can multiply $\frac{10}{3}$ by $2$ to get $\frac{20}{6}$.

    $\frac{11}{6} + \frac{20}{6} = \frac{31}{6} = 5 \frac{1}{6}$

  • Can you solve the addition problems?

    Hints

    You can convert the mixed numbers to improper fractions to solve or you can add the whole numbers and add the fractions separately.

    For example, when solving $6 \frac{4}{5} + 4 \frac{4}{15}$ we can add $6 + 4 = 10$ and then add $\frac{4}{5} + \frac{4}{15}$.

    $\frac{4}{5} + \frac{4}{15} = \frac{12}{15} + \frac{4}{15} = \frac{16}{15} = 1 \frac{1}{15}$

    Add $10 + 1$ $\frac{1}{15} = 11 \frac{1}{15}$

    Solution
    • $6 \frac{4}{5} + 4 \frac{4}{15} = 11 \frac{1}{15}$
    ${}$
    • $6 \frac{13}{18} + 5 \frac{1}{9} = 11 \frac{5}{6}$
    ${}$
    • $8 \frac{7}{12} + 3 \frac{1}{4} = 11 \frac{5}{6}$
    ${}$
    • $2 \frac{1}{5} + 10 \frac{4}{10} = 12 \frac{3}{5}$
    ${}$
    • $7 \frac{2}{5} + 5 \frac{3}{15} = 12 \frac{3}{5}$
    ${}$
    • $2 \frac{1}{6} + 4 \frac{1}{18} = 6 \frac{2}{9}$