Math Vale

🍰 Fraction Isles · Fractions & Proportion

Decimals

Read, compare and compute with decimals, and convert between decimals and fractions.

In short

  • The decimal point marks the ones column; every step right divides the place value by ten.
  • To compare decimals, pad them with zeros so they have the same number of places — more digits does not mean a bigger number.
  • Adding and subtracting needs the decimal points lined up; multiplying does not, but you must count the decimal places at the end.
  • A terminating decimal is a fraction over 10, 100 or 1000, which is why converting between them is mostly reading the last column.

The columns keep going

Place value does not stop at the ones column. Each step to the left multiplies by ten; each step to the right divides by ten. The decimal point simply marks where the ones column ends.

  • tens, ones . tenths, hundredths, thousandths

So 3.472 means 3 ones + 4 tenths + 7 hundredths + 2 thousandths. Written as fractions that is 3 + 4/10 + 7/100 + 2/1000.

A zero is never decoration. In 3.072 the 0 holds the 7 in the hundredths column; delete it and you have 3.72, a completely different number.

Comparing: pad, then compare

Longer does not mean larger. 0.45 is less than 0.5, even though 45 is bigger than 5, because 0.45 is 4 tenths and a bit while 0.5 is a full 5 tenths.

The safe method is to give both numbers the same number of decimal places by writing zeros on the end. Writing 0.5 as 0.50 changes how it looks but not what it is worth, because 5 tenths and 50 hundredths are the same amount.

  • 0.45 becomes 0.45
  • 0.5 becomes 0.50
  • 45 hundredths is less than 50 hundredths, so 0.45 < 0.5

A number line helps too: mark 0 and 1, split the gap into ten, then split one of those tenths into ten again. Every extra decimal place zooms in by a factor of ten.

Adding and subtracting: line up the points

When you add or subtract, tenths must meet tenths and hundredths must meet hundredths. That happens automatically if you write the numbers one above the other with the decimal points in a vertical line, filling any short number with zeros.

2.50 + 0.75 ------ 3.25

Lining the numbers up by their last digit instead is the classic wrecking mistake: it would add 0.75 to 2.5 as though the 5 were hundredths.

Finish by bringing the decimal point straight down into the answer, then check against an estimate: 2.5 + 0.75 should be a bit over 3, and 3.25 is.

Multiplying and dividing: count and shift

Multiplying does not need the points lined up at all. Ignore them, multiply as whole numbers, then count how many decimal places the question contained altogether and put that many into the answer.

  • 0.4 x 0.3: work out 4 x 3 = 12, count 1 + 1 = 2 places, so the answer is 0.12.

That rule is not magic. 0.4 is 4/10 and 0.3 is 3/10, so the product is 12/100 — two zeros in the denominator, two decimal places.

Dividing by a decimal is awkward, so make the divisor whole. Multiplying *both* numbers by the same power of ten leaves the answer unchanged, exactly as 6/2 and 60/20 are both 3.

  • 4.8 / 0.4 becomes 48 / 4 = 12

Decimals and fractions are the same idea

A decimal is just a fraction whose denominator is a power of ten, so converting is mostly reading.

Going decimal to fraction: name the last column. 0.7 is 7 tenths, so 7/10. 0.28 is 28 hundredths, so 28/100 = 7/25 after simplifying.

Going fraction to decimal: either divide the top by the bottom, or scale the fraction up to a denominator of 10, 100 or 1000.

  • 3/8: scale up by 125 to get 375/1000 = 0.375
  • 2/5: scale up by 2 to get 4/10 = 0.4

Not every fraction terminates. 1/3 = 0.333... repeats forever, because 3 does not divide any power of ten. A fraction gives a terminating decimal exactly when its simplified denominator is built only from 2s and 5s.

Worked examples

Example 1

Order these from smallest to largest: 0.6, 0.55, 0.506

  1. Give every number three decimal places: 0.600, 0.550, 0.506.
  2. Now all three are counted in thousandths, so they can be compared directly.
  3. Compare the tenths first: 6, 5, 5. So 0.600 is the largest.
  4. The two remaining tie on tenths, so compare hundredths: 5 against 0. That makes 0.506 the smallest.
  5. Smallest to largest: 0.506, 0.55, 0.6.

Example 2

7.2 / 0.9 = ?

  1. The divisor 0.9 is a decimal, which makes the division awkward.
  2. Multiply both numbers by 10 so the divisor becomes a whole number: 72 / 9.
  3. This is allowed because scaling both parts of a division by the same amount leaves the answer unchanged.
  4. 72 / 9 = 8.
  5. Check by multiplying back: 8 x 0.9 = 7.2.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Write this number in digits: 3 ones + 6 tenths

Answer: 3.6

  1. Columns needed: 3 ones, 6 tenths.
  2. Whole part: 3 ones = 3.
  3. Decimal part: 6 tenths.
  4. Putting the columns together gives 3.6.

Problem 2

Difficulty 3 of 5

Which symbol makes this true? 76.4 __ 76.40

  1. <
  2. >
  3. =

Answer: C. =

  1. Give both numbers 2 decimal places: 76.40 and 76.40.
  2. Compare column by column from the left: whole part first, then tenths, then hundredths.
  3. So 76.4 = 76.40.

Problem 3

Difficulty 4 of 5

Aisha keeps the guild ledger in gold pieces. The morning entry reads 10.137 gold and the afternoon entry adds 296.25 gold. What does the ledger total?

Answer: 306.387 gold

  1. Line the decimal points up and pad with zeros: 10.137 + 296.250.
  2. Add column by column from the right.
  3. Bring the decimal point straight down: 306.387.
  4. Estimate check: 10 + 296 is about 306, so 306.387 is sensible.

Common mistakes

  • Judging size by digit count and claiming 0.45 > 0.5.
  • Lining numbers up by their last digit instead of by the decimal point when adding.
  • Losing a decimal place in a product: writing 0.4 x 0.3 = 1.2 instead of 0.12.
  • Dropping a zero placeholder, turning 3.072 into 3.72.
  • Writing 3/8 as 0.3 by copying the numerator after the point.

What you should be able to do

  • Compare and order decimals using place value.
  • Add, subtract, multiply and divide decimals.
  • Convert between decimals and fractions in both directions.
  • Round decimals and use them in money and measurement problems.

Where this fits in the curriculum

Common Core

  • 5.NBT.A.3

    Grade 5 — Read, write and compare decimals to thousandths using place value.

  • 5.NBT.B.7

    Grade 5 — Add, subtract, multiply and divide decimals to hundredths.

  • 6.NS.B.3

    Grade 6 — Fluently add, subtract, multiply and divide multi-digit decimals.

  • 7.NS.A.2.D

    Grade 7 — Convert a rational number to a decimal, and recognise terminating and repeating decimals.

Ontario

  • G5.B1.5

    Grade 5 — Read, represent, compare and order decimal numbers up to hundredths.

  • G6.B1.4

    Grade 6 — Read, represent, compare and order decimal numbers up to thousandths.

  • G7.B2.9

    Grade 7 — Multiply and divide decimal numbers by decimal numbers, in various contexts.

SAT

  • Problem Solving and Data Analysis

    Decimal computation inside rate, unit and measurement problems.

    Decimal arithmetic is assumed rather than assessed; it is tagged here because every rate and unit question is written in decimals.

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