1.10 Fractions and decimals

Evaluate $-\frac{7}{10}\div\frac{4}{9}$−710​÷​49​.

Think: Following the steps above, we want to keep the first fraction $-\frac{7}{10}$−710​ as is, change the division operation to multiplication, and finally take the reciprocal of (or flip) $\frac{4}{9}$49​. We also know that we are dividing a negative number by a positive number, so the result will be a negative number.

Do: The reciprocal of $\frac{4}{9}$49​ is $\frac{9}{4}$94​. We then want to multiply by this giving us $-\frac{7}{10}\div\frac{4}{9}=-\frac{7}{10}\times\frac{9}{4}$−710​÷​49​=−710​×94​.

We can now multiply the numerators and denominators, $-\frac{7\times9}{10\times4}$−7×910×4​ and evaluate the multiplications taking care to use the right signs, $-\frac{63}{40}$−6340​. So $-\frac{7}{10}\div\frac{4}{9}=-\frac{63}{40}$−710​÷​49​=−6340​.

Reflect: By combining all we know about rational numbers and directed numbers, we can follow a series of simple steps to arrive at the correct answer. We knew before calculating anything that our answer was going to be a negative number. It is useful to know whether to expect a positive or negative answer before evaluating, as this can help us know if we have made a mistake if we end up with an answer that is not the expected sign.

Evaluate $4.83\times\left(-5.7\right)$4.83×(−5.7)

Think: We are multiplying a positive number by a negative number, so we know that the product will be negative. Ignoring the signs we can now calculate $4.83\times5.7$4.83×5.7 as we normally would, remembering to add the negative sign back to our answer.

Do: Before we even begin to calculate the answer it can be a good idea to have an estimation of the answer, especially when dealing with decimals. This will help us confirm our final answer is of the right magnitude.

Rounding both numbers to the nearest whole, gives us the much simpler calculation $5\times6$5×6, which we can evaluate to get $30$30, so we would expect our answer to be close to this value, taking into account the negative, we can expect our answer to be roughly equal to $-30$−30.

Now, to start the process, we ignore the decimal points. In this case we get $483$483 and $57$57. We can then multiply these together using the vertical algorithm:

Now, to account for the decimal point, we add the total number of decimal places in the original numbers together.

In this case the original numbers are $4.83$4.83, which has two decimal places, and $5.7$5.7, which has one decimal place. So their product will have $2+1=3$2+1=3 decimal places.

Then to find the answer, we take the product that we calculated before and insert the decimal point such that there are $3$3 decimal places.

Finally, we need to account for the fact that we were actually multiplying by $-5.7$−5.7. So our final answer will be $-27.531$−27.531.

This is very close to our original estimate of $-30$−30.