Another Peek Beyond the Point · Lesson 4 of 13
Is the Product Always Greater?
“Predict how decimal factors affect product size and use powers of ten in practical calculations.”
• Compare a positive product with its factors without always calculating it. • Explain multiplication by numbers greater than one or between zero and one. • Use multiplication by 10, 100, and 1000 through place value. • Apply product reasoning to costs, profit, and measurement conversions.
Is the Product Always Greater?
With whole numbers greater than one, multiplication often makes an amount larger. Decimals show why that is not a universal rule. Multiplying 8 by 0.25 means taking a quarter of 8, which is 2. The result is less than 8 even though the operation is multiplication. Think about the size of the multiplier, not just the operation’s name.
For a positive starting amount, multiplying by a number greater than 1 increases it; multiplying by 1 keeps it unchanged; multiplying by a number between 0 and 1 decreases it. Multiplying by 0 gives 0. These observations concern positive quantities in this chapter. They also explain how a product compares with both factors when neither is zero.
| Positive factors | Relationship to both factors | Example |
|---|---|---|
| Both greater than 1 | Greater than both | 3.4 × 6.5 = 22.1 |
| Both between 0 and 1 | Less than both | 0.75 × 0.4 = 0.3 |
| One between 0 and 1; the other greater than 1 | Between the two factors | 0.75 × 5 = 3.75 |
| One factor is 1 | Equal to the other factor | 1 × 0.75 = 0.75 |
Problem
Is 0.25 × 0.8 greater than either factor?
- 1.Both factors lie between 0 and 1, so multiplying either by the other takes only part of it.
- 2.The product must therefore be less than both 0.25 and 0.8.
- 3.Calculating confirms this: 25 × 8 = 200 with three decimal places gives 0.200 = 0.2.
Be careful when comparing with 1. If both positive factors are below 1, their product is below 1. But when one factor is above 1 and one below, knowing only that arrangement does not determine whether the product is below 1. For instance, 7 × 0.6 = 4.2, while 0.7 × 0.6 = 0.42. Estimation or a short calculation settles the question.
Multiplying by powers of ten
Multiplication by 10 makes every digit’s value ten times as large. A tenth becomes a one, and a hundredth becomes a tenth. Multiplication by 100 makes this change twice, and multiplication by 1000 makes it three times. We describe the written shortcut as moving the decimal point right, but the underlying change is in the value of every digit.
| Starting number | × 10 | × 100 | × 1000 |
|---|---|---|---|
| 5.7 | 57 | 570 | 5700 |
| 23.02 | 230.2 | 2302 | 23020 |
| 0.306 | 3.06 | 30.6 | 306 |
Problem
Calculate 0.306 × 1000.
- 1.0.306 is 306/1000. Multiplying by 1000 gives (306/1000) × 1000 = 306.
- 2.Its 3 tenths become 3 hundreds, its 0 hundredths become 0 tens, and its 6 thousandths become 6 ones.
- 3.The result is 306, not 30.6. Every original digit’s value has been multiplied by 1000.
Apply the product and check its meaning
Multiplication may count how many equal pieces are needed, combine a price with a mass, or turn a profit per item into a total profit. Some applications also require converting the resulting unit. Identify the amount per item or per unit first, then multiply by the number of items or the quantity used. Predicting the answer’s size helps catch an incorrectly placed decimal point.
Problem
Each coin in a stack is 1.45 mm thick. Find the height of 36 coins in centimetres.
- 1.The stack height is 1.45 × 36 = 52.20 mm.
- 2.There are 10 mm in 1 cm, so 52.20 mm = 52.20 ÷ 10 = 5.22 cm.
- 3.A height of 52.2 cm would be ten times too large. The multiplication gives millimetres first; the later division changes the unit.
Problem
A notebook costs a seller ₹23.6 and is sold for ₹30. Find the profit on 50 notebooks.
- 1.The profit per notebook is the selling price minus the purchase price: 30 − 23.6 = ₹6.4.
- 2.The total profit is 6.4 × 50 = 6.4 × 5 × 10 = 32 × 10 = ₹320.
- 3.As another check, sales bring in ₹1500 and the notebooks cost ₹1180. Their difference is ₹320.
“Multiplication makes bigger” works only under suitable conditions. Also, a trailing zero changes no value: 2.250 kg and 2.25 kg represent the same mass, so either gives the same cost at a fixed price per kilogram.
Quiz
For positive a, what happens to a when multiplied by 0.4?
Which product is less than both its positive factors?
What is 23.02 × 100?
Which product is below 1?
What is the height in centimetres of 36 coins, each 1.45 mm thick?
Practice Problems
- Compare 0.8 × 0.7 with each factor before finding its value.
- Explain why 0.75 × 5 lies between 0.75 and 5.
- Complete multiplication-by-10, 100, and 1000 tables for 0.92 and 24.67.
- Oranges cost ₹56.50 per kg. Find the price of 2.250 kg and check it against the costs of 2 kg and 3 kg.
- Find a product with one factor above 1 and one below 1 whose answer is less than 1. Find another whose answer exceeds 1.
- An item gives a profit of ₹4.75 per sale. Find the profit from 24 sales, showing your steps.
- Explain why multiplying a metre measurement by 100 gives its numerical value in centimetres.
Key Takeaways
• For positive amounts, a multiplier below 1 reduces the starting amount. • A multiplier above 1 increases it, and a multiplier of 1 preserves it. • Both factors matter when comparing a product with 1. • Multiplying by powers of ten increases digit values by the corresponding factor. • Practical problems require keeping track of units and intermediate quantities.