Question Video: Evaluating the Sum and Difference of Rational Numbers Mathematics • 7th Grade

Video Transcript

Evaluate two-fifths minus one-eighth plus 0.4 minus 0.025, giving the answer as a fraction in its simplest form.

In this question, we are asked to evaluate the sum and difference of four rational numbers, giving our answer as a fraction in its simplest form.

To answer this question, we can first recall that we can evaluate the addition and subtraction in any order. So, we can start by evaluating 0.4 minus 0.025. Evaluating this difference gives us two-fifths minus one-eighth plus 0.375. To add and subtract the remaining rational numbers, we will rewrite all of the numbers to be fractions. One way of doing this is to note that 0.375 is equal to 375 over 1,000. However, this is not the simplest form of this fraction.

By factoring or otherwise, we can note that 375 is equal to 125 times three and 1,000 is equal to 125 times eight. So they share a factor of 125. Canceling the shared factor of 125 in the numerator and denominator leaves us with three-eighths. So, we have two-fifths minus one-eighth plus three-eighths. To add or subtract fractions, we need their denominators to be equal. We can see in our expression that one-eighth and three-eighths have the same denominator. Therefore, we can add these fractions by adding their numerators.

We note that subtracting one-eighth is the same as adding negative one over eight. So we have two-fifths plus negative one plus three over eight. We can then evaluate this sum to obtain two-fifths plus two over eight. We can then simplify two-eighths by canceling the shared factor of two in the numerator and denominator to obtain one-quarter.

We now need to evaluate two-fifths plus one-quarter. To add these fractions together, we need them to have the same denominator. To rewrite the fractions so that they have the same denominator, we first find that the lowest common multiple of five and four is 20. So, we want to rewrite both fractions to have a denominator of 20. We can rewrite the fractions to have the same denominator by multiplying the numerator and denominator of the first fraction by four and the numerator and denominator of the second fraction by five. We can then evaluate each of the products to see that two-fifths is equal to eight over 20 and one-quarter is equal to five over 20.

Now that the fractions have the same denominator, we can add them together by adding their numerators. We have eight plus five over 20 is equal to 13 over 20. Finally, we see that 13 and 20 share no nontrivial common factors. So we cannot simplify any further. Hence, our answer is 13 over 20.