Question
Calculate and explain conceptually why dividing by a decimal less than 1 gives a result larger than the original number.
Solution — Step by Step
Multiply numerator and denominator by 10 to eliminate the decimal from the divisor:
Check: ✓
So .
Division asks: “How many times does the divisor fit into the dividend?”
— how many fives fit in 23.5? Only 4.7 times.
— how many halves fit in 23.5? Forty-seven times.
A smaller divisor fits more times. When the divisor is less than 1, it fits more times than the dividend itself, so the result is larger than the original number.
Why This Works
Think of it with a real-world analogy. You have 23.5 litres of juice. If each glass holds 5 litres, you can fill glasses. If each glass holds 0.5 litres (half a litre), you can fill glasses — many more, because each glass takes less.
The mathematical reason: dividing by is the same as multiplying by . When , then , so multiplication makes the number bigger.
Alternative Method — Converting to Fractions
Dividing by a fraction = multiplying by its reciprocal. This makes it clear why the result grows.
Common Mistake
Students sometimes think “dividing always makes smaller.” This is only true when dividing by a number greater than 1. Dividing by a number between 0 and 1 makes the result larger (e.g., , ). And dividing by 1 keeps the number the same.
A quick mental check: dividing by 0.5 is the same as multiplying by 2. Dividing by 0.25 is the same as multiplying by 4. Dividing by 0.1 is the same as multiplying by 10. Whenever you divide by a decimal, convert it to a fraction first, then multiply by the reciprocal.