Video: Finding the Rational Number Lying Halfway between Two Numbers

Find the rational number lying halfway between −2/7 and 4/35.

03:17

Video Transcript

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.

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