Comparing Decimals
At the shop one bag of apples weighs 0.7 kg and another weighs
0.65 kg — which is heavier? Picking the bigger amount means comparing the
decimals, and that comes up whenever you weigh things, check prices, or read a measuring scale.
We already know that
decimals count
smaller and smaller pieces of one — first tenths, then
hundredths. To decide which of two decimals is bigger, we use
the very same place value, working from the biggest pieces down.
The whole method is just three steps:
- Line up the decimal points — write one number under the
other so the dots sit in a column, tenths over tenths, hundredths over
hundredths.
- Compare from the LEFT — start at the biggest column
(the tenths) and walk rightwards.
- Stop at the first column that differs. The bigger digit
there wins for the whole number — nothing further right can change the answer.
So we compare the tenths first. The decimal with more tenths is
bigger — you can stop right there. Only if the tenths tie do you move on
to the hundredths.
0.7 > 0.65
Here 0.7 wins because it has seven tenths
while 0.65 has only six — and seven tenths
beats six tenths, no matter what comes after.
Pad with zeros to make them the same length
Lining up is easier when both decimals have the same number of digits
after the point. A decimal does not change if we add trailing
zeros on the right — they just say "and no more pieces". So
0.5 is exactly the same amount as
0.50:
0.5 = 0.50 = \tfrac{5}{10} = \tfrac{50}{100}
Now stack the pair you want to compare and pad the shorter one. To compare
0.5 and 0.45, write
0.5 as 0.50:
\begin{array}{c} 0.50 \\ 0.45 \end{array} \qquad 5 \text{ tenths} \;>\; 4 \text{ tenths} \;\Rightarrow\; 0.5 > 0.45
The tenths already decide it: 5 tenths beats
4 tenths, so 0.5 > 0.45.
We never even needed the hundredths.
Three worked examples
1) Tenths settle it straight away.
0.8 \;?\; 0.74 \;\Rightarrow\; 0.80 \text{ vs } 0.74 \;\Rightarrow\; 8 > 7 \;\Rightarrow\; 0.8 > 0.74
Eight tenths beats seven tenths. The extra 4 in
0.74 is only hundredths — far too small to catch up.
2) The tenths tie, so the hundredths decide.
0.83 \;?\; 0.87 \;\Rightarrow\; 8 = 8 \;\text{(tie)},\; \text{then } 3 < 7 \;\Rightarrow\; 0.83 < 0.87
Both have 8 tenths, so we walk one column right to the
hundredths: 3 against 7.
Seven wins, so 0.87 is bigger.
3) Whole numbers first of all.
3.4 \;?\; 3.38 \;\Rightarrow\; 3 = 3,\; \text{then } 4 > 3 \;\Rightarrow\; 3.4 > 3.38
The whole-number parts tie at 3, so we compare tenths:
4 tenths beats 3 tenths. Once
a column decides, you stop — 3.4 > 3.38.
The two decimal traps that catch everyone:
- Longer is NOT bigger. 0.5 > 0.45
even though 0.45 has more digits. With whole numbers
more digits does mean bigger; with decimals it does not. Count
the value of each column, never the number of digits.
- Compare tenths before hundredths. Always walk from the LEFT.
Never let a big hundredths digit (like the 5 in
0.45) trick you before you have checked the tenths.
See it: shaded tenths-strips
Each strip is one whole, cut into ten equal tenths. We shade in
the decimal — so 0.7 fills seven tenths and a bit more
shades part of the next cell. The longer shaded bar is the bigger
decimal, and we highlight its outline. Press Refresh for
a new pair to compare.
Watch out for the most common trap: more digits does not mean
bigger! 0.65 looks longer than
0.7, so it is tempting to call it the larger
number. But length is not value — only place value decides. Compare the tenths
first, and 0.7 clearly comes out ahead.
Press play, then replay it. Two decimals between
0 and 1 are plotted on
the number line; whichever sits further right is the bigger one, and
we say the comparison aloud.
Decimals in real life
Turtle clocked 9.4 seconds in the
race; Frog clocked 9.38 seconds. The
smaller time is the faster runner. The whole-second parts tie at
9, so compare tenths:
4 tenths against 3 tenths.
Frog's 9.38 is the smaller time, so
Frog won — even though 9.38 has more
digits!
9.38 s
<
9.4 s
Llama measures 1.5 metres and Goat
measures 1.45 metres. Pad the shorter
number: 1.5 becomes 1.50.
The wholes tie at 1, then the tenths decide —
5 tenths beats 4 tenths. So
1.5 > 1.45 and Llama is taller,
even though Goat's number looks longer.
1.5 m
>
1.45 m
One jar holds £0.8 and another holds
£0.75. Money is just decimals of a
pound, so the same rule works: 0.80 versus
0.75 — eight tenths beats seven tenths. The first jar
(£0.80, that is 80p) is worth more than
£0.75 (75p).
£0.8
>
£0.75
Khan Academy compares and orders decimals here: