Numbers like 10, 20, 30, 40, 50, 60, and so on are all multiples of 10. In multiples of ten, the ones place is always occupied by zero. To multiply two-digit numbers by multiples of ten, we apply place value and commutative properties. When you multiply a number by ten, it moves one position to the left on the place value chart.
Example: 10 × 10 = 100
20 × 10 = 200
30 × 10 = 300 and so on.
Follow these steps to multiply by multiple of ten:
- Multiply the digits of the numbers that aren’t zero.
- Count how many zeros there are in each factor.
- To the product, add the same number of zeros.
For example, To find, 50 × 20.
There are different methods to multiply by multiples of ten.
Method 1: By using place value
Take 50 × 20 = 5 × 2 × 100
= 1o × 100
So,
50 × 20 = 1000 (Multiply)
Therefore as explained earlier, when you multiply a number by ten, the digit of the number shifts to the left in the place value chart.
Method 2: By using associative property of multiplication
To “associate” with something means to join or connect with it. The product of three or more numbers, according to the associative property of multiplication, remains the same regardless of how the numbers are grouped.
50 × 20 = 50× (2 × 10) (Rewriting the 20 as 2 10)
= (50 × 2) × 10 (Using the associative property of multiplication)
= 100 × 10
= 1000
So, 50 × 20 = 1000.
For example, To find \(45\times10^{2}\).
Using the place value = 45 × 10 × 10
= 45 × 10 tens
= 45 × 100
= 4500
So, \(45\times10^{2}\) = 4500.
The number of zeroes in the answer is equal to the exponent 2 or the square of the number 10.
Thus, the product will have the same number of zeros as the exponent.
The number of zeroes in the product is equal to the number of zeroes in the factors.
When you find powers of a number in order, the list of products that results, forms a pattern. We can predict the next product on the list by looking at this pattern.
For example, To find 450 × 50.
= (45 × 10)(5 × 10) (Using place value)
= (45 × 5) × 10 × 10
= 225 × 100
= 22500
So, by using place value 450 × 50 = 22500.
Here, 450 has 1 zero and 50 has 1 zero. Therefore the product will have 1 + 1 = 2 zeroes, and hence the product is 22500.
For calculating the products of multiplication equations we can apply estimation to the operation. by rounding numbers to the nearest tens, hundreds, thousands, etc. When we estimate and multiply the number we find the approximate value compared to the exact answer.
Using the rounding method: We round the given factors to the required place value or to the nearest whole number and multiply in order to get our estimate and check to see how it compares to the exact answer.
Estimating products assist us in determining whether our response is reasonable. To calculate the product, round the multiplier and multiplicand to the nearest tens, hundreds, or thousands, then multiply the rounded numbers.
For example, To estimate 76 × 92.6.
Round to the nearest ten and then multiply.
Therefore, 76 × 92.6 = 80 × 92 (76 is rounded up and 92.6 is rounded down)
= 7360 (multiply)
So, 76 × 92.6 is about 7360.
For example, To estimate the product of 59 × 71.
Round to the nearest ten and then multiply.
Therefore, 59 × 71 = 60 × 70 (59 is rounded up and 71 is rounded down)
= 4200 (multiply)
So, 59 × 71 is about 4200.
Using compatible numbers: Compatibility numbers are those that are simple to multiply. Compatible numbers are close in value to the real numbers, making it easier to estimate the answer and solve problems. To make the numbers compatible, we can round them to the nearest tens, hundreds, thousands, or even ten thousands.
For example, To estimate the product of 76 × 92.6.
Choose compatible numbers and round to the nearest ten..
Therefore, 76 × 92.6 = 75 × 100 (76 is close to 75 and 92.6 is close to 100)
= 7500 (multiply)
So, 76 × 92.6 is about 7500.
While estimating the number, knowing whether our estimate is higher or lower than the actual value is extremely useful. It’s harder to tell whether an estimate is an overestimate or an underestimate when the factors are rounded up or when others are rounded down.
Overestimate values: An overestimate occurs when the estimate is higher than the actual value. The estimate is an overestimate if factors are only rounded up.
For example, To estimate 76 × 92.6.
Round to the nearest ten.
Therefore, 76 × 92.6 = 80 × 92 (76 is rounded up and 92.6 is rounded down)
= 7360 (multiply)
So, 76 × 92.6 is about 7360. The actual value is 7037. But the approximate product is 7360. An overestimate occurs when the estimate is higher than the actual product.
So, here the value is overestimated.
Underestimate values: An underestimate occurs when the estimate is lower than the actual product of the factors. The estimate is an underestimate if the factors are only rounded down.
For example, To estimate 76 × 92.6.
Round to the nearest ten.
Therefore, 76 × 92.6 = 75 × 90 (76 is rounded down as well as 92.6 being rounded down)
= 6750 (multiply)
So, 76 × 92.6 is about 6750. The actual product of the factors is 7037. But the approximate product is 6750.
So, here the value is underestimated.
1) Estimate 169.81 × 17.
Solution: We have to use rounding. And round to the nearest ten.
Therefore, 169.81 × 17 = 170 × 15 (170 is rounded up and 17 is rounded down)
= 2550 (multiply)
So, 169.81 × 17 is about 2550. The actual product is 2886. But the approximate product is 2550. An underestimate occurs when the estimate is lower than the actual product. So, here the product is underestimated.
2) Estimate 58 × 39.
Solution: We have to use compatible numbers. And round to the nearest ten.
Therefore, 58 × 39 = 60 × 40 (58 is close to 60 and 39 is close to 40)
= (6 × 4)× 100
= 24 × 100 (multiply)
= 2400
So, 58 × 39 is about 2400. The actual product is 2262. But the approximate product is 2400. An overestimate occurs when the estimate is higher than the actual product.
So, here the product is overestimated.
3) Estimate 54 × 23.
Solution: We have to use rounding. And round to the nearest ten.
Therefore, 54 × 23 = 60 × 20 (54 is rounded up and 23 is rounded down)
= 1200 (multiply)
So, 54 × 23 is about 1200. The actual product is 1242. But the approximate product is 1200.An underestimate occurs when the estimate is lower than the actual product.
So, here the product is underestimated.
4) A giant panda eats 32 pounds of food each day. An orca eats 18 times as much food as the panda eats each day. How much food does the orca eat each day?
Solution: Estimate 32 × 18.
We have to use rounding. And round to the nearest ten.
Therefore, 32 × 18 = 30 × 20 (32 is rounded down and 18 is rounded up)
= 600 (multiply)
So, 32 ×18 is about 600. The actual product is 576. And the approximate quantity of food that the orca eats is 600 pounds. This shows us that there is an overestimation as the estimate is higher than the actual quantity. So, Here the value is overestimated.
Therefore, the orca eats 576 pounds of food per day.
5) One can contains 1.26 liters of mango juice. You purchase a case of nine cans. How much mango juice do you purchase?
Solution: We have to estimate 1.26 × 9.
Round to the nearest ten = 1 × 10 (1.26 is rounded down and 10 is rounded up)
= 10 liters of mango juice
So, 1.26 × 9 is about 10. The actual quantity of juice is 11. But the approximated quantity is 10. An underestimate is seen in this situation as the estimate is lower than the actual quantity.
So, here the quantity is underestimated.
6) On a motorcycle, a gallon of gas will get you 51.2 miles. A car can travel 28.8 miles on one gallon of gasoline. How far can a motorcycle with 7 gallons of gas travel compared to a car with the same amount of gas?
Solution:
Distance traveled by the motorcycle with 7 gallons of gas = 51.2 × 7
Estimate 51.2 × 7.
Use compatible numbers and round to the nearest ten.
Therefore, 50 × 10 = (5 × 1) × 100 (51.27 is close to 50 and 7 is close to 10)
= 500 miles
So, 51.2 7 is about 500 miles.
Distance traveled by the car with 7 gallons of gas = 28.8 x 7
Estimate 28.8 x 7.
We have to use compatible numbers, once again. And round to the nearest ten.
Therefore, 30 × 10 = (3 × 1) × 2 tens (28.8 is close to 30 and 7 is close to 10)
= 300 miles
So, 28.8 × 7 is about 300 miles.
Comparing the distances traveled, 500 – 300 = 200 miles
Therefore, a motorcycle with 7 gallons of gas travels approximately 200 miles more on 7 gallons of gas when compared to a car with the same quantity of gas.
7) In a greenhouse, there are 509 new plants. A worker instructs a robot to arrange the plants in 17 rows, each with 20 plants. How many plants will the rows be unable to accommodate?
Solution: If the robot arranged the 20 plants in 17 rows,
Then this gives us, 20 × 17
= (2 × 17) × 10 (Using the commutative property of multiplication)
= 34 × 10
= 340
Now, the number of plants that didn’t fit in the rows will be,
= 509 – 340
= 169 plants.
Follow these steps to multiply by a multiple of ten:
- Multiply the digits of the numbers that aren’t zero.
- Count how many zeros there are in each factor.
- Add the same number of zeros to the product.
We round the given factors to the required place value or to the nearest whole number and multiply in order to get our estimated product and check to see how it compares to the exact answer. To calculate the product, round the multiplier and multiplicand to the nearest tens, hundreds, or thousands, then multiply the rounded numbers.
Compatibility numbers are those that are simple to multiply. Compatible numbers are close in value to the real numbers, making it easier to estimate the answer and solve problems.
An overestimate occurs when the estimate is higher than the actual product of the factors. An underestimate occurs when the estimate is lower than the actual product.