Mason plays guitar 1.2 hours a day. If he played for a total of 10.8 hours, how many days did he play for?
The first piece of information we’re given in this problem is the fact that Mason plays guitar for 1.2 hours every day. The question then goes on to ask us if he played for a total of 10.8 hours, how many days did he play for? In other words, how many lots of 1.2 can we fit in 10.8? This is a division question. To find the answer, we need to divide 10.8 by 1.2.
At the moment, we’re dividing a decimal by another decimal. And this is quite tricky to do. It would be much easier if you’re working with two whole numbers. What can we do to 10.8 to turn it into a whole number? The quickest way to turn 10.8 into a whole number is to shift all three digits one place to the left. One ten would become 100. Zero ones would become zero tens. And eight tens would move into the ones place. And what have we done to our number to cause the digits to shift like this? Each digit is now worth 10 times as much as it did before. We’ve multiplied 10.8 by 10. This is what happens when we multiply numbers by 10. They shift one place to the left.
We’ve turned our decimal into a whole number. This is gonna make it a lot easier to deal with. But we’re still dividing by a decimal, 1.2. If we want the answer in our division to stay the same, we’re going to have to multiply this number by 10 too. And again, both digits are going to shift one place to the left. One one will become one ten and two tens will become two ones. 1.2 multiplied by 10 gives us an answer of 12. And so, our division has become 108 divided by 12. And we’ll work out the answer in a second.
But is the answer we’re going to get the answer we’re looking for? Because we’ve altered these numbers. We’ve multiplied them both by 10. Well, the answer to that question is that the result of the division is going to be exactly the same. Perhaps, the best way to understand why this is is to think of our divisions as fractions. Remember that a fraction is just a division, one number divided by another. So although it might look a bit strange because it’s working with decimals, we could write our first division as 10.8 over 1.2. All this means is 10.8 divided by 1.2. It’s just another way of writing it.
Now, what do we know about equivalent fractions? We know that if we want two fractions to have the same value, if we multiply the numerator by one number, we need to also multiply the denominator by the same number. And so, if we multiply 10.8 by 10 to turn it into a whole number, we’re also going to need to multiply 1.2 by the same amount. And this, of course, will turn it into a whole number. This is how we know that the answer to 108 divided by 12 is going to be exactly the same as 10.8 divided by 1.2. So let’s get on with finding our answer.
How many 12s are there in 108? Well, a fact that we can use to help us is 10 12ths. 10 lots of 12 are 120. This is greater than 108. But all we have to do is subtract one lot of 12 from 120. And this will give us our answer of 108. Instead of 10 12ths, this is equal to nine 12ths. And so, we can say that 108 divided by 12 equals nine. And as we’ve proved already, this is exactly the same answer as if we divided our two decimals together.
Mason plays guitar 1.2 hours a day. If he played for a total of 10.8 hours, Mason would have played for nine days.