Lesson 50: Decimal Number Line (Tenths)

New Concept

To divide a number by a fraction, find the fraction's reciprocal. Then, multiply the original number by that reciprocal to find the answer.

To solve a problem like $7 \div \frac{2}{3}$:

Step 1: Find the number of $\frac{2}{3}$s in 1.

Property

On a number line, the distance between consecutive whole numbers can be divided into ten equal lengths. Each length is $\frac{1}{10}$ of the distance between consecutive whole numbers.

Examples

An arrow points to the eighth mark after the whole number 3, which represents $3\frac{8}{10}$ or 3.8.
An arrow points to the second mark after the whole number 9, which represents $9\frac{2}{10}$ or 9.2.
An arrow points to the fifth mark after the whole number 6, which represents $6\frac{5}{10}$ or 6.5.

Explanation

Imagine a ruler where the space between whole numbers is chopped into 10 equal bits called tenths. Each tiny bit is worth $\frac{1}{10}$, or 0.1. So, if you start at the number 5 and hop over 4 of these tiny marks, you land on $5\frac{4}{10}$. We just write this in its cool decimal form: 5.4! It’s that simple.

Property

To divide a whole number by a fraction, first find how many parts of that fraction are in 1 (its reciprocal), then multiply that result by the whole number.

Examples

To solve $8 \div \frac{2}{3}$: Step 1 is $1 \div \frac{2}{3} = \frac{3}{2}$. Step 2 is $8 \times \frac{3}{2} = 12$.
To solve $10 \div \frac{1}{5}$: Step 1 is $1 \div \frac{1}{5} = 5$. Step 2 is $10 \times 5 = 50$.
To solve $12 \div \frac{3}{8}$: Step 1 is $1 \div \frac{3}{8} = \frac{8}{3}$. Step 2 is $12 \times \frac{8}{3} = 32$.

Explanation

Dividing by a fraction sounds scary, but it’s a two-step trick! First, ask how many of the fraction fit into the number 1—this is just its reciprocal. Then, multiply that answer by the big number you started with. This turns a confusing division problem into a simple multiplication, making it way easier to find your answer and avoid a fraction-induced headache.

Property

Two numbers whose product is 1 are reciprocals. For example, $\frac{1}{2} \times 2 = 1$, so $\frac{1}{2}$ and $2$ are reciprocals.

Examples

The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$, because $\frac{5}{8} \times \frac{8}{5} = \frac{40}{40} = 1$.
The reciprocal of the whole number 7 (or $\frac{7}{1}$) is $\frac{1}{7}$, because $7 \times \frac{1}{7} = 1$.
The reciprocal of $\frac{1}{10}$ is 10 (or $\frac{10}{1}$), because $\frac{1}{10} \times 10 = 1$.

Explanation

Reciprocals are math’s dynamic duo! They are two numbers that, when multiplied, give you a perfect 1. To find a fraction's reciprocal, you just flip it upside down. For example, $\frac{2}{5}$ becomes $\frac{5}{2}$. This 'flipping' trick is the secret weapon that lets you transform tricky fraction division problems into much friendlier multiplication tasks. It is very useful.