Mason spent his free time watching TV and playing video games. If he spent two and two-fifth hours watching TV and two and five twelfth hours playing video games, how much free time did he have?
We know that the two activities that Mason does in his spare time are watching TV and playing video games. And we’re told how long Mason spends doing this. The length of time he spends watching TV is two and two-fifth hours. And the length of time that he spends playing video games is two and five twelfths hours. So we’re asked, how much free time did Mason have? In other words, what’s the total of two and two-fifth hours and two and five twelfths hours?
What could we say about these numbers? They’re both mixed numbers. This means they contain both a whole number part and also a fraction part. So to add two mixed numbers together, we first could add the whole number parts then the fraction parts. And we could combine the two at the end to give us our answer. So let’s begin by adding our whole number parts together. Two hours plus two hours equals four hours.
Now we need to add the fraction parts. What’s two-fifths plus five twelfths? Well, if we look at these fractions closely, we can see that they have different denominators. We have a number of fifths and a number of twelfths. In order for us to be able to add these two fractions, we need to convert them so that they have the same denominator, a common denominator.
What number is both a multiple of five and also 12? You can sometimes identify a common multiple quickly just by looking at the two numbers. But when we have two numbers like these that require a little bit of thought, sometimes the quickest way to find a common multiple is just to multiply the two numbers together. Five multiplied by 10 equals 50. So five multiplied by 12 must be another two lots of five more than 50. That’s another 10. And so a number that is both a multiple of five and 12 is 10 more than 50, or 60.
We can convert both our fractions so that they have 60 as their denominator. As we’ve just said, five multiplied by 12 equals 60. And to keep the fraction being worth the same, we need to also multiply the numerator by 12 too. Two lots of 12 are 24. Two-fifths is the same as twenty-four sixtieths.
The denominator in our second fraction is 12 of course. And to convert this into a denominator of 60, we need to multiply it by five. So that our fraction is still worth the same, we need to also multiply the numerator by five. Five fives are 25. And so we can say five twelfths is the same as twenty-five sixtieths.
Now that both our fractions have a common denominator, we can add them together. How many sixtieths are the same as twenty-four sixtieths plus twenty-five sixtieths? Well, we know double 24 is 48. So 24 plus 25 is one more than 48. The answer is forty-nine sixtieths. If we look quickly at the numbers 49 and 60, we can see that there doesn’t seem to be a number that we can divide both numbers by so that we could simplify it. This fraction is gonna have to stay as it is.
Now let’s go back to the addition that we were trying to work out. If you remember, we were trying to add the two fractions parts of our mixed numbers, two-fifths plus five twelfths. And we can now say that the answer is forty-nine sixtieths. So to find the overall total, we now need to add the whole number total to the fraction total.
Four plus forty-nine sixtieths is going to give us a mixed number. And it’s easy to see what this number is going to be, four and forty-nine sixtieths. If Mason’s free time is spent watching TV for two and two-fifth hours and playing video games for two and five twelfth hours, the total amount of free time that he has is four and forty-nine sixtieth hours.