Liam and Elizabeth both started cycling along a track in the same direction. Liam cycled two-fifths of a mile and then stopped. Elizabeth cycled four-fifths of a mile and then turned around and cycled back along the path for two-thirds of a mile. Who is further from the starting point? And what is the distance between them?
Liam cycled two-fifths of a mile. Elizabeth went four-fifths of a mile in one direction. And then she turned around and cycled for two-thirds of a mile in the opposite direction. If she cycled four-fifths of a mile in one direction and then turned around for two-thirds of a mile and we want to know how far she was from the starting point, we need to use subtraction. We need to subtract two-thirds of a mile from the four-fifths that she went in the beginning.
In order to subtract fractions, we need a common denominator. The least common denominator from three and five is 15. Five times three equals 15. And if we multiply the denominator by three, we need to multiply the numerator by three. Four-fifths is the same thing as twelve fifteenths. And for the two-thirds, what times three equals 15? Times five. And that means we also multiply the numerator by five. Two times five equals 10. Twelve fifteenths minus ten fifteenths. We subtract the numerator. 12 minus 10 equals two. And the denominator stays the same. Elizabeth is two fifteenths of a mile from where she started.
If we want to compare two-fifths to two fifteenths, we can find a common denominator between the two. To write two-fifths as a fraction over 15, the denominator has been multiplied by three. We can do the same to the numerator. Two times three equals six. And that means Liam has gone six fifteenths from the starting point.
Here is Liam at six fifteenths or two-fifths and Elizabeth at two fifteenths or four-fifths. And now Elizabeth is at two fifteenths of a mile from her starting point. Our first question wants to know who is farther from the starting point. Liam is farther from the starting point. If Elizabeth is two fifteenths of a mile from the starting point and Liam is six fifteenths of a mile from the starting point, there’s a distance of four fifteenths between them. This is because six fifteenths minus two fifteenths equals four fifteenths.
There was four fifteenths of a mile between the two of them.