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

Video Transcript

Evaluate one-third minus five twelfths plus three-eighths minus one-sixth, 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 all given as fractions. We need to give our answer as a fraction in its simplest form. To answer this question, we can begin by recalling that we can add and subtract fractions by first rewriting them to have the same denominator. In general, we have that 𝑎 over 𝑐 plus 𝑏 over 𝑐 is equal to 𝑎 plus 𝑏 over 𝑐. And 𝑎 over 𝑐 minus 𝑏 over 𝑐 is equal to 𝑎 minus 𝑏 over 𝑐. In other words, we can add and subtract the numerators of fractions when they have the same denominator.

This result holds for the sum and difference of any number of fractions, provided that they all have the same denominator. Therefore, we can evaluate the expression by first rewriting all four fractions to have the same denominator. To rewrite each of the fractions to have the same denominator, we need to find the lowest common multiple of the four denominators. We can start by noting that 12 is the lowest common multiple of three and 12. We can then consider the lowest common multiple of 12 and eight. We see that this is 24.

Finally, we can note that 24 is a multiple of six, so the lowest common multiple of the four denominators is 24. We want to rewrite each fraction to have a denominator of 24. To do this, we need to multiply the numerator and denominator of each fraction by the same value. First, we multiply the numerator and denominator of the first fraction by eight. Next, we multiply the numerator and denominator of the second fraction by two. Then, we multiply the numerator and denominator of the third fraction by three. Finally, we multiply the numerator and denominator of the fourth fraction by four to obtain the following expression.

We can now evaluate each of the products. The denominators should all evaluate to give 24. We have eight over 24 minus 10 over 24 plus nine over 24 minus four over 24. Now that the denominators are all the same, we can add and subtract the numerators of the fractions to get eight minus 10 plus nine minus four all over 24. Evaluating the numerator then gives us three over 24. Remember, we need to give our answer in its simplest form. We can note that the numerator and denominator are both divisible by three. Canceling this shared factor of three gives us one-eighth, which is our final answer.