Another Peek Beyond the Point · Lesson 2 of 13
Decimal Multiplication
“Use repeated addition and fractions to understand products involving decimal quantities.”
• Interpret a decimal multiplied by a whole number as repeated equal amounts. • Calculate decimal products through equivalent fractions. • Choose multiplication in cost, distance, and area situations. • Keep units and multi-step reasoning clear.
Decimal Multiplication
If one pen costs ₹9.50, five identical pens cost five equal amounts of ₹9.50. Multiplication combines those equal amounts. We can add 9.5 five times, just as we would add a whole number repeatedly. The fact that the amount is written as a decimal does not change the meaning of the operation.
Another way to calculate is to name the amount in smaller units. 9.5 is 95 tenths. Five lots of 95 tenths make 475 tenths, which is 47.5. This idea connects the familiar whole-number multiplication 95 × 5 with the decimal answer. We must still remember what unit those 475 pieces represent.
The numbers being multiplied are factors; the result is their product. In 9.5 × 5, 9.5 is the multiplicand, the amount being multiplied, and 5 is the multiplier. Interchanging the two factors leaves the product unchanged.
Problem
A pen costs ₹9.5. Find the cost of five pens.
- 1.Five pens cost 9.5 + 9.5 + 9.5 + 9.5 + 9.5 rupees.
- 2.Write 9.5 = 95/10. Then 9.5 × 5 = (95 × 5)/10 = 475/10.
- 3.475 tenths equal 47.5, so the cost is ₹47.50.
Multiply fractions representing the quantities
When a decimal is written as a fraction, we can use the fraction multiplication rule: multiply the numerators and multiply the denominators. A whole number can also be written as a fraction with denominator 1. This gives a reliable method for products involving one decimal or two decimals, without guessing where the point belongs.
Problem
Ajay’s home is 827 m from school. He walks there and back on six days. Find his weekly distance in kilometres.
- 1.There are 1000 m in a kilometre, so one journey is 827/1000 = 0.827 km.
- 2.Two journeys per day give 0.827 × 2 = 1654/1000 = 1.654 km.
- 3.Six days give 1.654 × 6 = 9924/1000 = 9.924 km.
- 4.Alternatively, 2 × 6 = 12 journeys, so 827 × 12 = 9924 m = 9.924 km. Both methods describe the same travel.
A decimal multiplier such as 7.5 cannot simply mean “add this amount a whole-number count of times”. It can mean seven and a half equal amounts. Fraction multiplication handles that situation naturally. For example, 12.5 km per litre multiplied by 7.5 litres gives a distance. The litres describe how many full or partial litre amounts are used.
Problem
A car travels 12.5 km per litre. How far does it travel using 7.5 litres?
- 1.The distance is 12.5 × 7.5 kilometres.
- 2.Write 12.5 = 125/10 and 7.5 = 75/10.
- 3.Multiply: (125 × 75)/(10 × 10) = 9375/100 = 93.75.
- 4.The distance is 93.75 km. As a check, 7 litres would give 87.5 km and 8 litres would give 100 km, so the answer lies between them.
Choose the operation and combine costs
A word problem does not always contain just one multiplication. When different items have different prices, calculate each group’s cost separately and then add. When the question asks for a total length or distance, count how many equal pieces or journeys there are before multiplying. Writing what each intermediate answer means helps prevent a correct calculation being used for the wrong quantity.
Problem
Meenu buys four notebooks at ₹15.50 each and three erasers at ₹2.75 each. Find her total spending.
- 1.The notebooks cost 15.50 × 4 = ₹62.00.
- 2.The erasers cost 2.75 × 3 = ₹8.25.
- 3.Add the two group costs: 62.00 + 8.25 = ₹70.25.
- 4.Multiplying 15.50 + 2.75 by either 4 or 3 would incorrectly assume equal quantities of both items.
Area uses multiplication too. A rectangle’s length times its breadth counts square units covering it. Decimal side lengths can be handled through the same fraction method. You will practise that calculation and a faster decimal-place rule in the next lesson. The reason for the multiplication remains the rectangle’s area relationship, not the appearance of decimals in the question.
Do not remove the decimal point and keep the whole-number product as the answer. 95 × 5 = 475, but 9.5 × 5 counts tenths and equals 47.5.
Quiz
Which expression gives the cost of six items at ₹2.5 each?
What is 7 × 0.3?
What does 0.827 × 2 represent in the walking example?
What is the denominator in (125/10) × (75/10)?
What is the cost of three erasers at ₹2.75 each?
Practice Problems
- Find 6 × 4 tenths and express the answer as a decimal.
- Calculate 9 × 5 hundredths, explaining the unit being counted.
- A shirt needs 1.65 m of cloth. Find the cloth required for three shirts.
- Calculate 27.34 × 6 by converting 27.34 into a fraction.
- A student travels 0.85 km each way for five days. Find the total distance.
- Use fractions to evaluate 4.23 × 3.7, then explain why the answer has the size it does.
- Make a shopping problem containing two item groups. Show the separate group costs and the final total.
Key Takeaways
• A decimal multiplied by a whole number combines equal decimal amounts. • Writing decimals as fractions gives meaning to the multiplication. • Multiply both numerators and denominators when using fractions. • Multi-step problems require identifying what each intermediate result represents. • A sensible estimate and the correct unit help check the answer.