David is trying to calculate one-fifth plus two-thirds. He has written the following as his working: one-fifth plus two-thirds is equal to one plus two over five plus three which is equal to three-eighths. His answer of three-eighths is wrong. Part a) Explain one mistake that David has made.
Well, where do we start with this one? Remember we can easily add, for example, one-fifth and two-fifths because the denominators are the same. We know we’re adding things that are the same size. In this question, the denominators are different. What we need to do is find a common denominator and then create equivalent fractions.
In this case, the lowest common multiple of three and five is 15 since that’s the smallest number in both the three and the five times tables. So the common denominator we’re going to choose is 15.
To get from five to 15, we multiply it by three. So we have to do the same to the numerator. One multiplied by three is three. So one-fifth is equivalent to three fifteenths. To get from three to 15, we multiply it by five. So once again, we have to multiply the numerator by five. Two multiplied by five is 10. So two-thirds is equivalent to ten fifteenths.
Now that we have these values, we can add the fractions, but we never add the denominators. What we’re really saying is that we have three fifteenths and then we add 10 more fifteenths and that gives us a total of thirteen fifteenths.
So in fact, David has made two mistakes. Firstly, he has not found a common denominator. Secondly, he added his denominators which we never do. This would not have happened had he found a common denominator in the first place.
Jenny is trying to calculate seven twelfths divided by five-thirds. She has written the following: seven twelfths divided by five-thirds is equal to twelve sevenths multiplied by three-fifths which is equal to 36 over 35. Her answer of 36 over 35 is wrong. Part b) Explain one mistake that Jenny has made.
Now, the first step here does at first glance look correct. To divide by a fraction, we change the division to a multiplication and then we find the reciprocal of the second fraction.
The reciprocal of five-thirds is indeed three-fifths since we can find it by swapping the numerator with the denominator. However, Jenny has also found the reciprocal of the first fraction which is incorrect. In fact, we leave the first fraction as it is and then we multiply as normal.
We multiply the numerator of the first fraction by the numerator of the second to give us 21 and then we multiply the two denominators: 12 multiplied by five is 60. The answer to this problem then is 21 over 60 which in fact simplifies as seven twentieths.
The mistake that Jenny has made is she found the reciprocal of the first fraction.