Arithmetic with rational numbers FAQ

When we add or subtract decimals, we need to line up the decimal points to make sure we are adding or subtracting the same place values. For example, if we want to add $3.4$‍ and $0.52$‍, we need to write them like this:

Notice how the decimal point is in the same column for both numbers. This helps us see that we are adding $4$‍ tenths to $5$‍ tenths, and $3$‍ ones to $0$‍ ones. If we don't line up the decimal points, we might get confused and add the wrong digits. For example, if we write them like this:

We might think that we are adding $4$‍ tenths to $2$‍ tenths, and $3$‍ ones to $5$‍ ones, which would give us the wrong answer of $8.6$‍. So, lining up the decimal points helps us avoid mistakes and add or subtract decimals correctly.

Sometimes, when we divide whole numbers, we get a remainder that is not zero. For example, if we divide $7$‍ by $2$‍, we get $3$‍ as the quotient and $1$‍ as the remainder. We write it like this:

$7÷2=3$‍ remainder $1$‍

But what if we want to know the exact answer, without a remainder? We can use decimals to show the part of the dividend that is left over after dividing by the divisor. We do this by adding a decimal point and a zero to the dividend, and continuing to divide using place value. For example, we can write $7$‍ as $7.0$‍ like this:

Now we can divide the $1.0$‍ by the $2$‍, and get $0.5$‍ (think $10$‍ tenths divided by $2$‍ is $5$‍ tenths). So in all, we got a quotient of $3+0.5$‍, which equals $3.5$‍. We write it like this:

We can do this with any whole number division that has a remainder. For example, if we divide $9$‍ by $4$‍, we get $2$‍ as the quotient and $1$‍ as the remainder. We write it like this:

$9÷4=2$‍ remainder $1$‍

To get the exact answer, we add a decimal point and two zeros to the dividend. We write it like this:

Now we can divide the $1.00$‍ by the $4$‍, and get $0.25$‍ (think $100$‍ hundredths divided by $4$‍ is $25$‍ hundredths). So in all, we got a quotient of $2+0.25$‍, which equals $2.25$‍. We write it like this:

To divide decimals by decimals, we need to make the divisor a whole number first. We do this by moving the decimal point in both the divisor and the dividend the same number of times to the right, until the divisor has no decimal point. This works because we are creating equivalent fractions.

For example, if we want to divide $3.6$‍ by $0.4$‍, we move the decimal point one time to the right in both numbers, and get $36÷4$‍. We write it like this:

Then we can divide as usual, and get $9$‍ as the answer. We write it like this:

Another example is $0.18÷0.06$‍. We move the decimal point two times to the right in both numbers, and get $18÷6$‍. We write it like this:

Then we can divide as usual, and get $3$‍ as the answer. We write it like this:

The reciprocal of a number answers the question, "How many groups of that number are in 1?"

Let's start with some basic numbers. How many groups of ${\displaystyle \frac{1}{3}}$‍ are in $1$‍? There are $3$‍ groups of ${\displaystyle \frac{1}{3}}$‍ in $1$‍, so $3$‍ (or ${\displaystyle \frac{3}{1}}$‍) is the reciprocal of ${\displaystyle \frac{1}{3}}$‍.

How many groups of $3$‍ are in $1$‍? There are no whole groups of $3$‍ in $1$‍, but there is a partial group of $3$‍ in $1$‍. Specifically, there is ${\displaystyle \frac{1}{3}}$‍ of a group of $3$‍ in the number $1$‍.

Here are some of the ways this pair of reciprocal values relates.

This pattern is true for all pairs of reciprocal numbers. Their product is always $1$‍.

Let's use that fact to find the reciprocal of ${\displaystyle \frac{5}{2}}$‍.

So the reciprocal of ${\displaystyle \frac{5}{2}}$‍ is ${\displaystyle \frac{2}{5}}$‍ and vice versa.

Notice a pattern? If we write a number in fraction form, then its reciprocal is a fraction with the numerator and the denominator flipped.

We often think of division as telling us how many groups of equal size we can make out of the total.

Suppose we want to know $5÷{\displaystyle \frac{2}{3}}$‍.

The reciprocal of a number answers the question, "How many groups of that number are in 1?" So the value of $1÷{\displaystyle \frac{2}{3}}$‍ will be the reciprocal of ${\displaystyle \frac{2}{3}}$‍, which ${\displaystyle \frac{3}{2}}$‍.

Instead, we want to know $5÷{\displaystyle \frac{2}{3}}$‍, which is $5$‍ times as much. So the quotient will also be $5$‍ times as much as $1÷{\displaystyle \frac{2}{3}}$‍.

Thus, $5\times {\displaystyle \frac{3}{2}}$‍ has the same value as $5÷{\displaystyle \frac{2}{3}}$‍.

There are many situations in real life where we might need to perform operations on fractions and decimals. For example, when we're cooking, we might need to multiply or divide fractions when measuring ingredients. If a recipe calls for ${\displaystyle \frac{3}{4}}$‍ of a cup of flour, but we only have a ${\displaystyle \frac{1}{2}}$‍ cup measure, we would need to divide ${\displaystyle \frac{3}{4}}÷{\displaystyle \frac{1}{2}}$‍ to know how many times to fill the measure to get the right amount.

We also encounter decimals when we're dealing with money. We might need to add or subtract amounts to calculate a total bill or to make change. For example, if we're buying two items that cost $\mathrm{\$}2.59$‍ and $\mathrm{\$}3.99$‍, we need to add $2.59+3.99$‍ to get a total of $\mathrm{\$}6.58$‍.

There are countless other situations where we might use fractions or decimals in everyday life – when we're cutting things into portions, calculating percentages, or dealing with measurements, just to name a few.