Comparing and ordering decimals

Is $0.3$ bigger than $0.27$? Many students wrongly say "$0.27$ is bigger because $27$ is bigger than $3$." Let's fix this for good.

Idea

To compare two decimals:

1. Equalise the digits after the point. Pad shorter numbers with trailing zeros , they don't change value. So compare $0.3$ as $0.30$ against $0.27$.
2. Compare the whole-number parts first. Bigger whole-part wins. ($2.5 > 1.9$.)
3. If whole parts are equal, compare digit by digit from the tenths place. First column with different digits decides.

With this rule, $0.30$ vs $0.27$: tenths digits are $3$ vs $2$. $3 > 2$, so $0.30 > 0.27$ , i.e., $0.3 > 0.27$.

A second helpful technique is the number line. Between $0$ and $1$, divide into $10$ equal parts; each tick is a tenth. Then $0.3$ is at the third tick and $0.27$ is between the second and third (closer to the third). $0.3$ sits to the right, so it is larger.

When ordering many decimals, line them all up by the decimal point and pad to the same length. Then sort column by column from left to right.

Worked examples

Example 1. Compare $0.5$ and $0.45$.

Pad: $0.50$ vs $0.45$. Tenths: $5 > 4$. So $0.5 > 0.45$.

Example 2. Compare $3.7$ and $3.700$.

Pad both to three decimal places: $3.700$ vs $3.700$. Equal. So $3.7 = 3.700$.

Example 3. Order from smallest to largest: $0.6, 0.06, 0.66, 0.606$.

Pad to three places: $0.600$, $0.060$, $0.660$, $0.606$. Compare: $0.060 < 0.600 < 0.606 < 0.660,$ i.e., $0.06 < 0.6 < 0.606 < 0.66$.

Example 4. A student ran $100$ m in $13.05$ s; another in $13.5$ s. Who is faster?

Pad: $13.05$ vs $13.50$. Tenths: $0 < 5$. So $13.05 < 13.50$. The first runner took less time , so the first runner is faster.

Try it yourself

1. Which is greater, $0.8$ or $0.79$?
2. Compare $2.05$ and $2.5$.
3. Order: $0.4, 0.04, 0.404, 0.044$ from smallest to largest.
4. Fill in $<, >, =$: $1.20 \, \square \, 1.2$.
5. Order: $0.999, 1.0, 0.9, 1.01$.
6. Two pencils cost $\rupee 7.5$ and $\rupee 7.05$. Which is cheaper?
7. Compare $\tfrac{3}{4}$ and $0.7$. (Convert to decimals first.)
8. Write a number that lies between $0.4$ and $0.5$.

Activity

Mark $10$ ticks between $0$ and $1$ on a long strip of paper. Place the decimals from question 3 above (or any list of your choice) at the correct ticks. Visualising helps cement the digit-by-digit rule.