Find the rational number lying
halfway between negative two-sevenths and four thirty-fifths.
Here, we have two fractions,
negative two-sevenths and four thirty-fifths. And we’re asked to find the
rational number which lies halfway between. We can recall the rational number
is a number which can be expressed as 𝑝 over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞
is not equal to zero. The best way to start a question
like this is to see if we can make the denominators the same value.
We should be able to write negative
two-sevenths as a fraction over 35. Observing that we multiply the
denominator by five, then our numerator will also be multiplied by five, which gives
a value of negative 10 over 35. It may help if we could visualize
negative 10 over 35 and four over 35 on a number line. So at the lower end of this number
line, we have got negative 10 over 35. And at the top end, we have four
thirty-fifths. This value of zero thirty-fifths
would also just be equivalent to zero.
If we were to think in terms of the
distance from zero, on the left-hand side, we have a distance of ten thirty-fifths
and on the right-hand side, a distance of four thirty-fifths. That’s equivalent to fourteen
thirty-fifths in total. Half of that would give us seven
thirty-fifths. Therefore, if we start from
negative 10 and count one, two, three, four, five, six, seven, we would get a value
of negative three thirty-fifths. As a check, counting down seven
thirty-fifths from four thirty-fifths would also give us negative three
thirty-fifths. Therefore, we could give the answer
as negative three thirty-fifths.
There is, of course, a different
method if we don’t want to or couldn’t draw it on a number line. Let’s go back to our two original
values: negative two-sevenths, which we could write as negative ten thirty-fifths,
and four thirty-fifths. The halfway point between these two
is equivalent to finding the median. We would begin by adding these two
fractions. To add fractions, we must have the
same denominator and we add the values on the numerator. In this case, negative 10 plus four
would give us negative six. So, remember, we’re finding the
median. So, we’ve added our values. And then, as there’s two values, we
need to divide by two.
In order to divide this fraction by
two, we can consider it as the fraction two over one, and then we multiply by the
reciprocal. So, we need to work out negative
six thirty-fifths multiplied by one-half. Before we multiply, we can simplify
this by taking out the common factor of two. So, we’ll have negative three
thirty-fifths multiplied by one over one. Multiplying the numerators and then
multiplying the denominators separately, we get the value of negative three
thirty-fifths, which confirms our earlier answer given by the first method.