Question Video: Evaluating the Difference between Two Rational Numbers Mathematics • 7th Grade

Video Transcript

Evaluate 0.65 minus one-fifth, giving the answer as a fraction in its simplest form.

In this question, we are asked to find the difference between two rational numbers, one given as a decimal and the other given as a fraction. We need to give our answer as a fraction in its simplest form. To find the difference between rational numbers, it is a good idea to write them in the same form. This means that one way that we can answer this question is by rewriting both numbers as decimals. We know that one-fifth is 0.2. So, this difference is equal to 0.65 minus 0.2. We can then evaluate this difference to get 0.45. We need to write this as a fraction in its simplest form.

To do this, we can start by writing the decimal as a fraction by noting that it is equal to 45 over 100. We can then cancel the shared factor of five in the numerator and denominator to obtain nine over 20. This cannot be simplified further, so it is our final answer.

While this method works for this particular question, in general, fractions may have difficult decimal expansions to add or subtract. So, in general, it is a good idea to convert both numbers into fractions rather than decimals.

To do this, let’s clear some space and then rewrite 0.65 as a fraction. We can note that it is equal to 65 over 100. We can then cancel the shared factor of five in the numerator and denominator to get 13 over 20. Therefore, the difference is equal to 13 over 20 minus one-fifth. We can then recall that we can find the difference between two fractions if they have the same denominator by finding the difference in their numerators. In general, we have 𝑎 over 𝑐 minus 𝑏 over 𝑐 is equal to 𝑎 minus 𝑏 all over 𝑐.

We want to rewrite both of our fractions to have the same denominator. To do this, we need to find the lowest common multiple of the denominators. In this case, 20 is a multiple of five, so the lowest common multiple is 20. The first fraction already has a denominator of 20. We can find an equivalent fraction to one-fifth with a denominator of 20 by multiplying the numerator and denominator by four. Evaluating the products then gives us 13 over 20 minus four over 20.

Now that the denominators of the fractions are equal, we can find their difference by finding the difference in their numerators. We get 13 minus four all over 20. We can then evaluate to obtain nine over 20. There are no nontrivial shared factors between the numerator and denominator, so we cannot simplify the fraction any further.