Multiplying and dividing decimals: count the places, move the point

Your child's grade 5 homework says 2.4 × 1.5, the answer key says 3.6, and the method you remember starts with lining up the decimal points. That method belongs to addition. Learning how to multiply decimals and how to divide decimals comes down to two habits: multiply as if the points were not there and count the places, and before you divide, make the divisor a whole number.

Why the decimal point moves when you multiply decimals

The decimal point moves because a decimal is a fraction with a denominator of 10, 100 or 1,000, and multiplying fractions multiplies the denominators. Take 0.3 × 0.2: that is 3/10 × 2/10 = 6/100, and 6/100 is written 0.06.

The whole-number part of the work is 3 × 2 = 6. The denominators decide where the point goes: tenths times tenths gives hundredths, so the answer needs two decimal places. If what a tenth or a hundredth is still feels shaky, start with the guide to decimals for parents, which covers reading, comparing and converting decimals.

Common Core standard 5.NBT.B.7 asks fifth graders to "add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value". A hundredths grid is the classic drawing.

The area model on a hundredths grid

The area model shows 0.3 × 0.2 as the overlap of two shaded strips on a 10 × 10 grid. Shade 3 of the 10 columns for 0.3 and 2 of the 10 rows for 0.2; the strips overlap in 3 × 2 = 6 small squares.

Each small square is 1/100 of the grid, so the overlap is 6/100 = 0.06. The picture also shows why the product is smaller than both factors: you are taking a part of a part.

How to multiply decimals, step by step

Multiplying decimals uses the whole-number algorithm with one extra step at the end: count the decimal places and place the point. The same five steps work for any size of number, including money.

1. Estimate first. Round each factor to a whole number you can multiply in your head: 2.4 × 1.5 is close to 2 × 2 = 4.
2. Ignore the decimal points. Write the numbers lined up on the right and multiply as whole numbers: 24 × 15 = 360.
3. Count the decimal places in both factors. 2.4 has one decimal place and 1.5 has one, so the product needs 1 + 1 = 2.
4. Place the point. Count two places in from the right of 360 to get 3.60, which equals 3.6.
5. Compare with the estimate. 3.6 is close to 4, so the point is in the right place; 36 or 0.36 would fail the check.

The same steps handle 0.25 × 0.4. The whole-number product is 25 × 4 = 100, the factors have 2 + 1 = 3 decimal places, and 100 with three decimal places is 0.100, which is 0.1. For the whole-number algorithm itself, the guide on how to teach long multiplication walks through it digit by digit.

Free printable worksheetMultiplication — 5th GradersWorksheet · PDF

The part people get wrong: placeholder zeros

Placeholder zeros are the zeros you write in front of a short product so it can have enough decimal places. For 0.03 × 0.2, the whole-number product is 3 × 2 = 6, and the factors have 2 + 1 = 3 decimal places.

The number 6 has only one digit, so write two zeros in front of it: 0.006. Skipping that step gives 0.6 or 0.06, which are 100 and 10 times too large.

How to divide decimals, step by step

Dividing decimals has two cases: dividing a decimal by a whole number, and dividing by a decimal. The second case turns into the first after one move.

Divide a decimal by a whole number. Divide as usual and put the decimal point in the quotient straight above the point in the dividend. For 8.4 ÷ 4, 8 ÷ 4 = 2, the point goes up, and 4 ÷ 4 = 1, so the answer is 2.1.

Divide by a decimal. Multiply the divisor and the dividend by the same power of 10 until the divisor is whole. For 4.2 ÷ 0.6, multiply both by 10 to get 42 ÷ 6 = 7; for 7.5 ÷ 0.25, multiply both by 100 to get 750 ÷ 25 = 30.

The OpenStax Prealgebra textbook states the rule in one line: "make the divisor a whole number by moving the decimal point all the way to the right", then move the point in the dividend "the same number of places to the right, writing zeros as needed". A whole number divided by a decimal follows it too: 12 ÷ 0.4 becomes 120 ÷ 4 = 30.

How to do long division with decimals

Long division with decimals is ordinary long division with the point carried up and zeros added to the dividend when a remainder is left. For 13.5 ÷ 6, the 6 goes into 13 twice with 1 left, the point goes up, and 6 goes into 15 twice with 3 left.

Write 13.5 as 13.50 and bring down the zero: 6 goes into 30 exactly 5 times, so 13.5 ÷ 6 = 2.25. The guide to teaching long division covers the divide, multiply, subtract, bring-down cycle that this step relies on.

Count the places vs fractions vs powers of ten, with money examples

The common decimal methods all give the same answer, but each one suits a different moment. Pick the method that fits the numbers in front of you, and keep the estimate as the check on every one.

Money and measurements are where these methods show up outside homework, and each example uses the same steps.

• Notebooks: 3 notebooks at $2.35 each is 235 × 3 = 705 with two decimal places, so $7.05.
• Gas: 12.5 gallons at $3.40 a gallon is 125 × 340 = 42,500 with three decimal places, so 42.500, which is $42.50.
• Walking: 3.2 miles a day for 5 days is 32 × 5 = 160 with one decimal place, so 16 miles.
• Ribbon: a 22.5-inch ribbon cut into 1.5-inch pieces is 22.5 ÷ 1.5 = 225 ÷ 15 = 15 pieces.
• Mileage: 261 miles on 7.25 gallons is 26,100 ÷ 725 = 36 miles per gallon.

A spoken estimate keeps the decimal point honest: "about 12 gallons at about $3 is about $36" makes $425 or $4.25 obviously wrong. Percent problems use the same multiplication, since 15% of $40 is 0.15 × 40 = 6, and the guide on how to calculate percentages builds on that step.

Multiplying and dividing by 10, 100 and 1,000

Multiplying by 10, 100 or 1,000 shifts each digit one, two or three places to the left, because each place is worth 10 times the place to its right. So 3.45 × 10 = 34.5, 3.45 × 100 = 345 and 3.45 × 1,000 = 3,450.

Dividing shifts the digits right: 3.45 ÷ 10 = 0.345 and 45 ÷ 100 = 0.45. Standard 5.NBT.A.2 asks children to "explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10", and the place value guide for parents explains why each place is ten times the next.

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How to check progress with decimal multiplication and division

Five short questions show whether the procedures are secure. Ask for the estimate first and the working second.

• Tenths times tenths: 0.6 × 0.7. The answer is 0.42, because 6 × 7 = 42 and the factors have two decimal places.
• A placeholder zero: 1.2 × 0.05. The answer is 0.06, from 12 × 5 = 60 with three places, which gives 0.060.
• Division by a decimal: 5.6 ÷ 0.8. The answer is 7, from 56 ÷ 8.
• A power of ten: 4.5 ÷ 100. The answer is 0.045.
• A whole number by a decimal: 9 ÷ 0.3. The answer is 30, from 90 ÷ 3, and it is larger than 9.

A workable test by grade

A grade 5 child should handle decimals to hundredths, and a grade 6 child should handle longer decimals with the written method. Standard 6.NS.B.3 asks sixth graders to "fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation".

England's national curriculum for mathematics sets a similar bar in Year 6: "multiply one-digit numbers with up to 2 decimal places by whole numbers" and "use written division methods in cases where the answer has up to 2 decimal places". For printable practice at that level, the division worksheet for 5th graders comes with an answer key.

Common mistakes with multiplying and dividing decimals

Four mistakes cause most wrong answers, and each has a one-line fix.

• Lining up the points to multiply: Lining up decimal points is the rule for adding, and bringing the point straight down in 2.4 × 1.5 gives 36.0. Line the numbers up on the right and count decimal places instead.
• Forgetting placeholder zeros: 0.03 × 0.2 written as 0.06 drops a place. The product needs three decimal places, so it is 0.006.
• Moving the point in only the divisor: Turning 7.5 ÷ 0.25 into 7.5 ÷ 25 gives 0.3, which is 100 times too small. Multiply both numbers by the same power of 10.
• Expecting multiplication to make numbers bigger: 0.5 × 8 = 4, which is smaller than 8, and 6 ÷ 0.5 = 12, which is larger than 6. Multiplying by a number below 1 shrinks the result, and dividing by it grows the result.

Edge case: zeros at the end of a product

Zeros at the end of a product count while you place the point and can be dropped afterwards. In 0.25 × 0.4, the product 100 needs three decimal places, giving 0.100, and that equals 0.1.

Dropping the zeros before counting is the trap: treat 100 as 1 and you get 0.001, which is 100 times too small. With money, keep two decimal places in the final answer, so $42.500 is written $42.50.

Your decimal multiplication and division checklist

• Estimate: round to whole numbers and say the rough answer before you start.
• Multiply: ignore the points, multiply as whole numbers, then count the decimal places in both factors.
• Zeros: write placeholder zeros in front of a short product; drop trailing zeros only after the point is placed.
• Divide: make the divisor whole by multiplying both numbers by 10, 100 or 1,000.
• Point up: in long division, the point in the quotient sits straight above the point in the dividend.
• Check: multiply the quotient by the divisor, or compare the product with your estimate.

Parents of fifth graders will find grade-level practice on the math game for 5th graders page.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice, in 33 languages. Twenty-two of its games play free in the browser with no sign-up, and 30 printable worksheets for grades 1–6 come with answer keys.