Question Video: Evaluating Algebraic Fractions by Substitution Mathematics • 6th Grade

Evaluate 7𝑞/(𝑞 + 2(𝑝 + 5)) for 𝑝 = 3 and 𝑞 = 12.

03:51

Video Transcript

Evaluate seven 𝑞 over 𝑞 plus two times 𝑝 plus five for 𝑝 is equal to three and 𝑞 is equal to 12.

In this question, we are asked to evaluate a given expression in terms of 𝑝 and 𝑞 using the given values of 𝑝 and 𝑞. To answer this question, the first thing we need to do is substitute 𝑝 equals three and 𝑞 equals 12 into the given expression. We obtain seven times 12 over 12 plus two times three plus five.

We now need to evaluate this expression. And to do this, we will recall two things. First, we recall that the acronym PEMDAS can help us recall the order of operations. We start by evaluating expressions inside parentheses; then, we move on to evaluating the exponents. Next, we evaluate multiplication, then division. After this, we evaluate any additions. And finally, we evaluate the subtractions.

It is worth noting that the stages of multiplication and division can be done in any order. Similarly, the stages of addition and subtraction can be done in either order. These pairs of operations are often considered in the same step.

There is one extra thing that we should note before we begin evaluating. That is that the fraction notation means that we evaluate the numerator and denominator separately and leave the division until last. We can think of this as adding parentheses over the numerator and denominator. However, we usually leave these out since it is implied by the fraction notation.

We are now ready to start evaluating the expression. We need to start with the expressions inside the parentheses. We see that only three plus five is in the parentheses, so we evaluate this to obtain eight. This gives us seven times 12 over 12 plus two times eight. We can now move on to evaluating any exponents. In this expression, we see that there are no exponents to evaluate.

So we will move onto the multiplication and division stage. Remember, we do not evaluate the division of the numerator and denominator until the end, since we are using fraction notation. We can evaluate the products in the numerator and denominator separately. We have that seven times 12 is equal to 84 and two times eight is equal to 16. This gives us 84 over 12 plus 16.

We can now move on to the final stage in the order of operations, addition and subtraction. We see that there is only one such operation, which is in the denominator of the fraction. We can calculate that 12 plus 16 is equal to 28. Therefore, we have 84 over 28.

Now that we have evaluated the expressions in the numerator and denominator, we can evaluate the division. One way of doing this is to note that 84 is equal to 28 times three. This means that 28 goes into 84 three times with no remainder. So 84 over 28 is equal to three.

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