Decimals & decimal place value
The idea
Place value does not stop at the ones — it keeps going to the right of a little dot called the decimal point. The first spot after the point counts tenths, and the next one counts hundredths. So 0.4 means 4 tenths, the same amount as the fraction 4/10, and 0.35 means 35 hundredths. You meet this system every day in money: a dime is one tenth of a dollar and a penny is one hundredth, so $2.35 is 2 dollars, 3 dimes, and 5 pennies.
Keep one rule front and center: digits only mean something through the place they occupy. The classic stumble is reading 0.35 as bigger than 0.4 because 35 looks bigger than 4. But 0.4 is 40 hundredths, and 40 beats 35. A handy fix is to give both numbers the same number of decimal places — write 0.4 as 0.40 — and then compare. Tacking a zero on the end changes nothing, because 4 tenths and 40 hundredths are the same amount, just counted in smaller pieces.
Worked example
Two paper frogs have a jumping contest. Priya's frog jumps 0.4 meters and Marco's frog jumps 0.35 meters. Whose frog jumped farther?
- It is tempting to say 35 beats 4, but those digits live in different places. The 4 in 0.4 sits in the tenths place, so it means 4 tenths of a meter.
- Rewrite 0.4 using hundredths so both jumps use the same size pieces: 4 tenths equals 40 hundredths, so 0.4 = 0.40.
- Now compare like with like: 40 hundredths is more than 35 hundredths, so 0.4 m is the longer jump.
- Money tells the same story: $0.40 is more than $0.35. Priya's frog wins by 0.40 − 0.35 = 0.05 m, which is 5 hundredths of a meter.
Answer. Priya's frog jumped farther: 0.4 m equals 0.40 m, which beats 0.35 m by 0.05 m.
Check your understanding
- Why does writing a zero on the end of 0.4 leave its value completely unchanged?
- How are dimes and pennies like the tenths and hundredths places of a dollar amount?
- What mistake is someone making when they read 0.35 as bigger than 0.4, and how would you straighten out their thinking?
- Where would 0.4 and 0.35 sit on a number line that runs from 0 to 1?