Video Transcript
If 𝑥 is equal to two and
one-third, 𝑦 is equal to three-quarters, and 𝑧 is equal to 0.7, evaluate 𝑥 times
𝑧 minus 𝑦.
In this question, we are asked to
evaluate an expression involving the product and difference of rational numbers. We could do this by using a
calculator. However, it is useful to be able to
solve these problems without a calculator.
When applying operations to
rational numbers, it is a good idea to have the numbers in the same form, since this
is the easiest way to combine them. In general, it is easiest to
multiply rational numbers written as fractions. So we can rewrite 𝑥 and 𝑧 as
fractions. We have that 𝑥 is equal to
seven-thirds and 𝑧 is equal to seven-tenths.
We can now substitute our values
for 𝑥, 𝑦, and 𝑧 into the expression to obtain seven-thirds times seven-tenths
minus three-quarters. This is the expression that we need
to evaluate. We can evaluate this expression by
first recalling that we evaluate the product before the subtraction and that we can
multiply two fractions by multiplying their numerators and denominators
separately. In general, we have that 𝑎 over 𝑏
times 𝑐 over 𝑑 is equal to 𝑎𝑐 over 𝑏𝑑.
Therefore, we can rewrite our
expression as seven times seven over three times 10 minus three-quarters. We can then evaluate the products
in the numerator and denominator to obtain 49 over 30 minus three-quarters.
To subtract two fractions, we need
their denominators to be equal. We can note that the lowest common
multiple of 30 and four is 60. We can rewrite both fractions to
have a denominator of 60. We multiply the numerator and
denominator of the first fraction by two and the numerator and denominator of the
second fraction by 15. Therefore, we have 98 over 60 minus
45 over 60.
Since the denominators of the two
fractions are now the same, we can subtract the numerators to get 53 over 60. We cannot simplify this fraction
since the greatest common factor of the numerator and denominator is one. So we leave our answer as 53 over
60.
