Ordering Decimals
Every question with a fully worked solution.
- Ordering Decimals - Video - Ordering Decimals from Smallest to Largest Watch
Theory
To compare decimals, give them the same number of decimal places, then compare the digits place by place from the left: the first place that is different decides. This Year 7 maths page shows how to read decimals on a number line, compare and order positive and negative decimals, find a decimal between two others and order fractions, decimals and percentages together.
A decimal is a fraction whose denominator is \(10\), \(100\), \(1000\), and so on, written in place-value form: \(\dfrac{25}{100} = 0.25\).
The digits after the decimal point are parts of a whole: tenths, hundredths and thousandths. In \(81.25\) the \(8\) is \(8\) tens, the \(1\) is \(1\) one, the \(2\) is \(2\) tenths and the \(5\) is \(5\) hundredths.
Zeros at the end of a decimal do not change it: \(0.4 = 0.40 = 0.400\). A zero straight after the point does matter: \(0.04\) is not \(0.4\).
Every positive number is larger than every negative number. For two negatives, the one with the bigger size is further left, so it is smaller: \(-0.8 < -0.75\).
Ascending order is smallest first. Descending order is largest first. In a race, the smallest time wins.
Fill with zeros, then compare from the left. The first different digit decides:
O = ones, t = tenths, h = hundredths, th = thousandths.
| Number | O | t | h | th |
|---|---|---|---|---|
| \(0.52\) | 0 | 5 | 2 | 0 |
| \(0.507\) | 0 | 5 | 0 | 7 |
How to compare two decimals
- Fill with zeros so both have the same number of decimal places.
- Compare the whole-number parts, then the tenths, then the hundredths, and so on.
- The first place that is different decides.
How to order a list of decimals
- Write them in a column with the points lined up, filling with zeros.
- Split into negatives and positives; every negative is smaller than every positive.
- Order each group, then join them (ascending: smallest first).
Write them in a column with three decimal places: \(1.070\), \(1.700\), \(1.077\), \(1.770\).
The ones are all \(1\). The tenths are \(0, 7, 0, 7\), so the smallest is \(1.070\) or \(1.077\). The hundredths are both \(7\); the thousandths are \(0 < 7\). The smallest is \(1.07\).
(ii) Find the smallest digit that makes this true: \(2.\square 4 > 2.58\)
(i) \(5.38 = 5.380\). Ones \(5 = 5\), tenths \(3 = 3\), hundredths \(0 < 8\).
(ii) The tenths digit decides first. \(\square = 5\) gives \(2.54 < 2.58\); \(\square = 6\) gives \(2.64 > 2.58\).
(i) \(5.308 < 5.38\)
(ii) \(6\)
Negatives, a bigger size first: \(-1.30, -1.25, -0.90\). Positives: \(0.05, 0.50\).
\(-1.3, -1.25, -0.9, 0.05, 0.5\)
Fill with zeros: \(-7.300\), \(-7.341\), \(-7.340\), \(7.300\). The positive is largest. For the negatives, the smallest size is closest to zero, so it comes next.
\(7.3, -7.3, -7.34, -7.341\)
| \(\dfrac{3}{5}\) | \(=\) | \(\dfrac{60}{100} = 0.60\) |
| \(62\%\) | \(=\) | \(0.62\) |
| \(\dfrac{13}{20}\) | \(=\) | \(\dfrac{65}{100} = 0.65\) |
\(0.65 > 0.62 > 0.60 > 0.58\), so 7D, 7C, 7A, 7B.
(ii) How many numbers with exactly two decimal places are between \(-0.5\) and \(-0.4\)?
(i) \(3.6 = 3.60\) and \(3.7 = 3.70\).
(ii) Between \(-0.50\) and \(-0.40\) are \(-0.49, -0.48, \ldots, -0.41\).
(i) \(3.65\)
(ii) \(9\) numbers
The fastest time is the smallest. Fill with zeros: \(11.90\), \(11.09\), \(11.19\), \(11.10\).
\(11.09\) s, \(11.1\) s, \(11.19\) s, \(11.9\) s
Common pitfalls
Frequently asked questions
How do you compare two decimals?
Give them the same number of decimal places, then compare from the left. \(0.52 = 0.520 > 0.507\).
Is 0.35 bigger than 0.4?
No. \(0.4 = 0.40\), and \(40\) hundredths is more than \(35\) hundredths.
Which is smaller, -0.8 or -0.75?
\(-0.8\). It is further below zero, so it is further left on the number line.
How do you find a decimal between two decimals?
Add a decimal place. Between \(2.7\) and \(2.8\) are \(2.71\) to \(2.79\).
How do you order fractions and decimals together?
Change them all to decimals first, then compare. \(\dfrac{3}{5} = 0.6\).