Evaluate 𝑥𝑦 plus 𝑧 if 𝑥 equals seven and three-fourths, 𝑦 equals nine and one-third, and 𝑧 equals four and three-sevenths.
The expression we’re trying to evaluate is 𝑥 times 𝑦 plus 𝑧. To solve this, we’ll need to substitute each of our values for 𝑥, 𝑦, and 𝑧 into our expression. However, the values for 𝑥, 𝑦, and 𝑧 are all given in mixed numbers. And to work with these values, we’ll want to find their equivalent improper fractions instead of the value of the mixed number.
Let’s start with 𝑥. If 𝑥 equals seven and three-fourths, to find its improper fraction, you multiply the whole number value by the denominator, seven times four. And then you add the value of the numerator. And all of this goes over the original denominator. We would have seven times four plus three over four. Seven times four is 28, plus three is 31. Seven and three-fourths can be rewritten as thirty-one fourths.
We need to follow the same procedure for 𝑦. The improper fraction will be nine times three plus one over three. Nine times three is 27, plus one is 28. Nine and one-thirds is equal to twenty-eight thirds. For 𝑧, four and three-sevenths, we have four times seven, which is 28, plus three, which is 31. And the denominator remains the same, seven. Four and three-sevenths is equal to thirty-one sevenths.
These improper fractions are the values that we’ll want to substitute into our expression. 𝑥 times 𝑦 becomes thirty-one fourths times twenty-eight thirds. And then the plus 𝑧 becomes plus thirty-one sevenths. When we multiply fractions, we’re multiplying their numerators and their denominators. But before we do that, we recognize that some things can cancel out. Four goes into 28 seven times, so we can simplify. Then we’ll have a numerator of 31 times seven over three in the denominator. And we’ll bring everything else down. We need to multiply 31 by seven. We know that 30 times seven would be 210, and one times seven is seven, which makes 31 times seven 217.
And we now need to add two hundred and seventeen thirds plus thirty-one sevenths. Remember, to add fractions, they need a common denominator. If we let the common denominator be 21, then we multiply two hundred and seventeen thirds by seven over seven. And we multiply 31 over seven by three over three. We can solve seven times 217 with partial products. 200 times seven is 1400. 10 times seven is 70. And seven times seven is 49.
To multiply 217 by seven, we would add the three partial products, which would give us 1519. We would have 1519 over 21. And then we’ll multiply 31 by three, which is 93. We have 93 over 21.
Now that we have a common denominator, we can add the numerators together, which is 1612 over 21. We haven’t been given instructions on how to simplify this value. But because all of the initial values were given as mixed numbers, we could calculate 1612 over 21 as a mixed number.
To find out what the mixed number would be, we want to calculate how many times 21 will go into 1612. At first, we might think, well, we could choose eight because 20 goes into 160 eight times. But because we’re dealing with 21, we should probably start with seven. We have seven times one is seven and seven times two is 14. If we subtract 147 from 161, we get 14, and we bring down the two.
Now we’re asking, how many times does 21 go into 142? We know that seven times 21 is 147, and that’s too much. So we should go with six. Six times one is six. Six times two is 12. And we’ll subtract 126 from 142, which gives us 16. Since 21 goes into 1612 76 times with the remainder of 16, we write the whole number as 76. And for the fraction piece of the mixed number, the remainder 16 is in the numerator and 21 is the denominator. 1612 over 21 written as a mixed number is 76 and 16 over 21.