Dividing Decimals by Moving the Decimal Point

A decimal divisor can make long division awkward. The standard strategy is to rewrite the problem with a whole-number divisor. You do this by multiplying both numbers in the division by the same power of 10. The quotient stays the same because multiplying the numerator and denominator of a fraction by the same nonzero number creates an equivalent fraction.

Why both numbers move together

Division can be written as a fraction: 4.8 ÷ 0.6 = 4.8/0.6. Multiplying both numerator and denominator by 10 gives 48/6, which equals 8. The original and rewritten expressions have the same value. If you moved only the divisor’s decimal point, you would change the denominator without making the corresponding change to the dividend, so you would change the answer.

Think of the operation as scaling both quantities equally. For 2.45 ÷ 0.07, multiply both values by 100: 245 ÷ 7. This is easier to calculate, and the quotient is 35. If you multiplied both by 10 instead, you would get 24.5 ÷ 0.7, which is still equal to 35 but keeps a decimal divisor. Choose enough places to make the divisor a whole number.

A step-by-step procedure

1. Count the number of digits after the decimal point in the divisor.
2. Multiply both the dividend and divisor by 10 for each place needed to make the divisor whole.
3. Rewrite the equivalent division problem, adding zeros to the dividend if the decimal shift requires them.
4. Divide as with whole numbers or place the decimal in the quotient over the decimal point in the rewritten dividend.
5. Check by multiplying the quotient by the original divisor; the product should equal the original dividend.

Worked examples

One decimal place in the divisor

Calculate 8.4 ÷ 0.7. The divisor has one decimal place, so multiply both numbers by 10: 84 ÷ 7 = 12. Check with the original values: 12 × 0.7 = 8.4. The quotient is 12.

Two decimal places in the divisor

Calculate 5.46 ÷ 0.13. Multiply both numbers by 100: 546 ÷ 13 = 42. Check: 42 × 0.13 = 5.46. The quotient is 42.

The dividend needs a zero

Calculate 3.6 ÷ 0.08. Move each decimal point two places to the right. The dividend 3.6 becomes 360, and the divisor 0.08 becomes 8. Then 360 ÷ 8 = 45. A zero is appended to 3.6 because its value can be written as 3.60 before shifting: 3.60 × 100 = 360. Check: 45 × 0.08 = 3.6.

A quotient less than 1

Calculate 0.84 ÷ 2.1. The divisor has one decimal place, so multiply both numbers by 10: 8.4 ÷ 21. Since 21 does not go into 8.4 once, the quotient is less than 1. Rewrite as 84/210 = 0.4, or divide 8.4 by 21 using long division. Check: 0.4 × 2.1 = 0.84.

Place the quotient’s decimal correctly

After making the divisor a whole number, the rewritten problem may still have a decimal dividend. For example, 7.5 ÷ 3 is solved by placing the quotient decimal above the decimal point in 7.5, giving 2.5. Another way to reason is that 3 × 2.5 = 7.5. Do not put the quotient decimal directly above a decimal point that belonged to the original divisor; it is the rewritten dividend that anchors the long-division placement.

If long division leaves a remainder, append zeros after the decimal point without changing the value. For 1 ÷ 8, write 1.000… and continue dividing to obtain 0.125. Use the precision requested by the problem. If the quotient is a repeating decimal, indicate repetition or round as directed.

Estimate before calculating

An estimate can catch a misplaced decimal. For 5.46 ÷ 0.13, round to 5.2 ÷ 0.13 = 40, so a quotient near 42 is plausible. For 3.6 ÷ 0.08, dividing by a positive number smaller than 1 should produce a result larger than 3.6; 45 is reasonable, while 0.45 would not be.

Use nearby compatible values. To estimate 7.9 ÷ 0.4, think of 8 ÷ 0.4 = 20. Since 7.9 is just below 8, the result should be just below 20. A calculated answer around 2 or 200 signals that the decimal shift may be wrong.

Interpret decimal division in context

Suppose 6.3 liters of juice are poured into bottles that hold 0.35 liter each. The number of bottles is 6.3 ÷ 0.35. Multiply both by 100 to get 630 ÷ 35 = 18 bottles. The units help clarify the operation: total liters divided by liters per bottle gives bottles.

In a money problem, a quotient may represent a number of purchases, a unit price, or a per-person share. Decide whether the answer should be whole or fractional. If a van holds 8 passengers and 45 people need transport, 45 ÷ 8 = 5.625 vans mathematically, but six whole vans are required. The arithmetic and the real-world rounding decision are separate steps.

Common errors and how to catch them

• Moving the decimal in the divisor but not the dividend. This changes the quotient.
• Moving the two decimal points different numbers of places. Apply the same power-of-10 factor to both values.
• Counting places in the dividend instead of in the divisor to decide the needed shift.
• Dropping a zero when the dividend has fewer decimal places than the divisor requires.
• Placing the decimal in the quotient based on the original divisor instead of the rewritten division problem.
• Assuming division always makes a number smaller. Dividing by a positive decimal less than 1 makes a positive dividend larger.
• Rounding a context-dependent count to the nearest integer when the situation requires rounding up.

Exam takeaway

Make the divisor whole by multiplying both the dividend and divisor by the same power of 10. Divide the equivalent numbers, estimate the expected scale, and verify with multiplication using the original divisor. Then interpret units and rounding from the context rather than treating the quotient as an isolated number.