Adding Mixed Numbers with Unlike Denominators

Adding Mixed Numbers with Unlike Denominators exercise

• 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

  • $6 \frac{4}{5} + 4 \frac{4}{15} = 11 \frac{1}{15}$

  ${}$

  • $6 \frac{13}{18} + 5 \frac{1}{9} = 11 \frac{5}{6}$

  ${}$

  • $8 \frac{7}{12} + 3 \frac{1}{4} = 11 \frac{5}{6}$

  ${}$

  • $2 \frac{1}{5} + 10 \frac{4}{10} = 12 \frac{3}{5}$

  ${}$

  • $7 \frac{2}{5} + 5 \frac{3}{15} = 12 \frac{3}{5}$

  ${}$

  • $2 \frac{1}{6} + 4 \frac{1}{18} = 6 \frac{2}{9}$