Chapter 1 Use Postitive Rational Numbers
Section 1.1: Fluently Add, Subtract, and Multiply Decimals
Page 7 Exercise 1 Answer
Maxine is connecting 4 cardboard tubes together vertically and each tube is 0.28 meter in length. The combined measure of the connected tubes is 4 times the length of the tubes:
Result
The combined measure of the connected tubes is 1.12 meter in length.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 7 Exercise 1 Answer
Maxine made another windmill model by connecting 4 cardboard tubes that are each 2.8 meters long.
The combined measure of this model is 11.2 meters.
In the other, similar problem, where Maxine made a windmill model by connecting 4 cardboard tubes, but the length of tubes was 0.28, the combined measure was 1.12. Notice that in this problem the decimal point was moved for one place. The length of tubes was 2.8 and the result was 11.2, the decimal point in the result also moved for one place.
For example, if we were to look at a similar problem, the only difference being that Maxine used tubes 28 meters long, we can assume that the answer would be 112. The decimal point was moved for one place in one of the factors, thus it is moved for one place in the result.
Result
The combined measure of this model is 11.2 meters. Compared to the other similar problem, the decimal point was moved one place for both the factor and the product.
Page 8 Exercise 1 Answer
Kim and Martin swam 50 meters. Martin took 0.47 seconds longer than Kim. Kim’s time was 50.9 seconds.
Martin’s time was 51.37 seconds.
In Martin finished the reace 0.267 seconds after Kim, the solution would be to add 0.267 to 50.9. Adding 0.267 to 50.9 is different from adding 0.26 to 50.9 for a tiny difference between 0.267 and 0.26
So, the result of 0.267 + 50.9 would be a little greater than 0.26 + 50.9, precisely it would be 0.007 points greater.
Result
Martin’s time was 51.37 seconds. Adding 0.267 to 50.9 is 0.007 greater than adding 0.26 to 50.9 since 0.267 is 0.007 greater than 0.26.
Page 9 Exercise 2 Answer
We know Amy finished the race in 20.7 seconds and Katie finish the race 0.13 seconds before Amy. To find Katie’s time in the race, we need to subtract 20.7 and 0.13.
To subtract, we first need to line up the decimals at their decimal points. Since 0.13 has two decimals places, we need to use a zero as a place holder for the 20.7.
We can’t subtract 3 from 0 se we will need to regroup in order to subtract the hundredths places:
Katie’s time in the race was then 20.57 seconds.
To use estimation to check that out answer is resonable, we can estimate the difference by rounding 0.13 to the same decimal place as 20.7. Rounding 0.13 to the nearest tenth gives 0.1 so 20.7 – 0.13 ≈ 20.7 – 0.1 = 20.6. Since 20.57 is close to the estimate of 20.6, then the answer is reasonable
Result
20.57 seconds
Page 9 Exercise 3 Answer
We multiplied the numbers using long multiplication method.
0.42 x 0.2 = 0.086
We can see that the first factor, 0.43, has two decimal places and the second factor, 0.2, has only one decimal place.
The result, 0.086, must have three decimal places because that is the sum of number of decimal places of the first and the second factor.
Result
0.43 has 2 decimal places, 0.2 has 1 decimal place, and the product has 3 decimal places.\
Page 10 Exercise 1 Answer
To add decimals, line up place values and add. Regroup as needed.
For example, add numbers 47.34 and 13.12.
To subtract decimals, line up place values and subtract. Regroup as needed.
For example, subtract 14.08 from 34.78.
To multiply decimals, multiply as you would with whole numbers. Then use the number of decimal places in the factors to place the decimal point in the product.
For example, multiply 7.98 and 21.09.
Result
To add and subtract decimals, line up place values and add or subtract. Regroup as needed. To multiply as you would with whole numbers. Then use the number of decimal places in the factors to place the decimal point in the product.
Page 10 Exercise 2 Answer
Adding or subtracting decimals is similar to adding and subtracting whole numbers in that both involve adding the like place values. However, when adding or subtracting decimals we must be careful to line up place values before adding or subtracting and use zeros as placeholders if necessary.
For example,
Result
Adding or subtracting decimals is similar to adding and subtracting whole numbers in that both involve adding the like place values. However, when adding or subtracting decimals we must be careful to line up place values before adding or subtracting and use zeros as placeholders if necessary.
Page 10 Exercise 3 Answer
If a decimal product has final zeros to the right of the decimal point, there is no need to write them.
For example, multiply 2.05 and 15.2.
2.05 x 15.2 = 31.160
The result is 31.160, but it is equal to write 31.16.
Result
If a decimal product has final zeros to the right of the decimal point, there is no need to write them.
Page 10 Exercise 4 Answer
Diego says that the product of 0.51 × 2.427 will have five decimal places.
Diego is right. One of the factors has two decimal points and the other three. Since, one ends with 1 and the other with 7, the last decimal place won’t be a zero, thus the product will have all five decimal places.
Let’s check.
0.51 x 2.427 = 1.23777
Diego was right, the product has five decimal places.
Result
Diego was right because the product is 1.23777, which has five decimal places.
Page 10 Exercise 5 Answer
We can add numbers using the long addition method.
Result
8.6
Page 10 Exercise 6 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Regroup if needed:
Result
1.06
Page 10 Exercise 7 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
Result
5.35
Page 10 Exercise 8 Answer
Subtract numbers using the long subtraction method. We use subscripts for carrying numbers.
9.62 – 0.3 = 9.32
Result
9.32
Page 10 Exercise 9 Answer
We can add numbers using the long addition method.
Result
10.276
Page 10 Exercise 10 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as place holders and regroup if needed:
Result
8.892
Page 10 Exercise 11 Answer
We need to place the decimal point in the correct location in the product 4 x 0.94 = 476
4 has 0 decimal places and 0.94 has 2 decimal places so the product must have 0 + 2 = 2 decimal places. The placement of the decimal point is then between the 3 and 7 to get 4 x 0.94 = 3.76
Result
4 x 0.94 = 3.76
Page 10 Exercise 12 Answer
We need to place the decimal point in the correct location in the product 5 x 0.487 = 2435.
5 has 0 decimal places and 0.487 has 3 decimal places so the product must have 0 + 3 = 3 decimal places. The placement of the decimal point is then between the 2 and 4 to get 5 x 0.487 = 2.435.
Result
5 × 0.487 = 2.435
Page 10 Exercise 13 Answer
We need to place the decimal point in the correct location in the product 3.4 × 6.8 = 2312
3.4 has 1 decimal place and 6.8 has 1 decimal place so the product must have 1 + 1 = 2 decimal places. The placement of the decimal point is then between the 3 and 1 to get 3.4 × 6.8 = 23.12
Result
3.4 × 6.8 = 23.12
Page 10 Exercise 14 Answer
We need to place the decimal point in the correct location in the product 3.9 × 0.08 = 312
3.9 has 1 decimal place and 0.08 has 2 decimal places so the product must have 1 + 2 = 3 decimal places. The placement of the decimal point is then to the left of the 3 to get 3.9 × 0.08 = 0.312
Result
3.9 x 0.08 = 0.312
Page 10 Exercise 15 Answer
We need to place the decimal point in the correct location in the product 0.9 × 0.22 = 198.
0.9 has 1 decimal place and 0.22 has 2 decimal places so the product must have 1 + 2 = 3 decimal places. The placement of the decimal point is then to the left of the 1 to get 0.9 × 0.22 = 0.198
Result
0.9 × 0.22 = 0.198
Page 10 Exercise 16 Answer
We need to place the decimal point in the correct location in the product 9 x 1.2 = 108.
9 has 0 decimal places and 1.2 has 1 decimal place so the product must have 0 + 1 = 1 decimal place. The placement of the decimal point is then between the 0 and 8 to get 9 × 1.2 = 10.8.
Result
9 × 1.2 = 10.8
Page 10 Exercise 17 Answer
Multiply numbers using long multiplication method.
5.3 x 2.7 = 14.31
Result
14.31
Page 10 Exercise 18 Answer
Multiply numbers using long multiplication method.
8 x 4.09 = 32.72
Result
32.72
Page 10 Exercise 21 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
Result
18.21
Page 11 Exercise 22 Answer
We can add numbers using the long addition method.
Result
6.985
Page 11 Exercise 23 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
Result
4.62
Page 11 Exercise 24 Answer
We can add numbers using the long addition method.
Result
27.185
Page 11 Exercise 25 Answer
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
Result
0.051
Page 11 Exercise 26 Answer
We can add numbers using the long addition method.
Result
23.8
Page 11 Exercise 27 Answer
We can add numbers using long addition method.
Result
26.89
Page 11 Exercise 28 Answer
Multiply numbers using long multiplication method.
7 x 0.5 = 3.5
Result
3.5
Page 11 Exercise 29 Answer
Multiply numbers using long multiplication method.
12 x 0.08
Result
0.96
Page 11 Exercise 30 Answer
Multiply numbers using long multiplication method.
24 x 0.17 = 4.08
Result
4.08
Page 11 Exercise 31 Answer
Multiply numbers using long multiplication method.
0.4 x 0.17 = 0.068
Result
0.068
Page 11 Exercise 32 Answer
Multiply numbers using long multiplication method.
1.9 x 0.46 = 0.874
Result
0.874
Page 11 Exercise 33 Answer
Multiply numbers using long multiplication method.
3.42 x 5.15 = 17.613
Result
17.613
Page 11 Exercise 34 Answer
Write an equation that illustrates the following:
A number with two decimal places multiplied by a number with one decimal place. The product has only two nonzero digits.
Since one of the numbers has two decimal places and the other one, the product will have three, so the last digit in the product must be a zero.
We’ll try with 0.12 and 1.5, hoping that 2 × 5 will give 10 “in the right” place.
0.12 x 1.5 = 0.180
The product is 0.180, but the last digit right of the decimal point is zero so the number can be written as 0.18.
The solution is the following equation.
0.12 × 1.5 = 0.18
Result
0.12 x 1.5 = 0.18
Page 11 Exercise 35 Answer
It is given that the entire shampoo bottle holds 6.35 ounces and that 1.078 ounces in the bottle is vanilla oil. We need to find how much of the bottle is NOT vanilla oil so we need to subtract 6.35 and 1.078.
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
Therefore, 5.272 ounces of the bottle is not vanilla oil.
Result
5.272 ounces
Page 11 Exercise 36 Answer
The fastest speed a table tennis ball has been hit is about 13.07 times as fast as the speed for the fastest swimming which is 5.35 as seen in the graph.
Multiply numbers using long multiplication method.
13.07 x 5.35 = 69.9245
The fastest speed a table tennis ball has been hit is 69.9245 miles per hour.
Result
69.9245 mph
Page 11 Exercise 37 Answer
The graph on the right shows Fastest Sporting Speeds – fastest swimming time is 5.35 miles per hour, fastest running is 27.9 mph, fastest rowing is 13.99 mph, fastest luge is 95.69 mph, and fastest thrown baseball 106 mph.
How fast would 1.5 times the fastest rowing speed be?
The number of decimal places in the answer will be the sum of numbers of decimal places of 1.5 and of the time of the fastest rowing speed. The fastest rowing speed is 13.99 mph, so the sum is equal to 3. The number of decimal places in the answer be 3.
1.5 x 13.99 = 20.985
1.5 times the fastest rowing speed is 20.985.
Result
20.985
Page 11 Exercise 38 Answer
The graph on the right shows Fastest Sporting Speeds – fastest swimming time is 5.35 miles per hour, fastest running is 27.9 mph, fastest rowing is 13.99 mph, fastest luge is 95.69 mph, and fastest thrown baseball 106 mph.
Which activity has a recorded speed about 7 times as fast as the fastest rowing speed?
7 x 13,99 = 97.93
Number closest to 97.93, out Fastest Sporting Speeds, is 95.69, so the answer is fastest luge.
Result
Fastest Luge.
Page 12 Exercise 39 Answer
Matthew bought a jersey, a pennant, and a hat, paid with a 50 bill and some money he borrowed from his friend, and got $6.01 in change from the cashier. We need to find how much he borrowed from his friend to pay for all the items.
The bill shows the following – the price of the jersey $39.99, the pennant $ 10.25, and the had $13.75.
Matthew paid $ 63.99 for the jersey, the pennant and the hat.
Matthew gave the cashier exactly the amount of the bill plus the change she has given him back.
Matthew gave the cashier $70. His friend let him borrow the difference between the money he gave the cashier and the 50 bill.
Matthew borrowed $20 from his friend.
Result
Matthew borrowed $20 from his friend.
Page 12 Exercise 40 Answer
It is given that Anna’s running time was 23.1 seconds and another runner’s time was 5.86 seconds faster. To fin the other runner’s time, we need to subtract 23.1 and 5.68.
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Write additional zeros as placeholders and regroup if needed:
The other runner’s time was then 17.24 seconds.
Result
17.24 seconds
Page 12 Exercise 41 Answer
The product of 0.25 × 0.4 has only one decimal place.
The first factor has two decimals places and the other has one, so the product should have three decimal places. However, when 25 and 4 are multiplied they give 100 which mean that the two last digits in the product will be zeros, hence there is no need to write them, and the product has only one decimal place.
0.25 x 0.4
Result
The last two digits in the product will be zeros so there is no need to write them.
Page 12 Exercise 42 Answer
The wings of some hummingbird’s beat 52 times per second when hovering. If a hummingbird hovers for 35.5 seconds, its wings beat 35.5 × 52.
35.5 x 52 = 1846
Result
1846
Page 12 Exercise 43 Answer
The students at Walden Middle School are selling tins of popcorn to raise money for new uniforms. One tin costs $9.25. They sold 42 tins in the first week.
9.25 x 42 = 388.5
Result
The students made $388.5 in the first week.
Page 12 Exercise 44a Answer
There are four trails in Joshua Tree National Park, they are: Lost Horse Mine which is 6.4 kilometers long, Lost Palms Oasis 11.6, Mastodon Peak 4.8, and Skull Rock 2.7 kilometers.
What is the combined length of the Lost Horse Mine trail and the Mastodon Peak trail?
To find the answer we need to calculate the sum of the length of the Lost Horst Mine trail and the Mastodon Peak trail
Result
The combined length of the Lost Horse Mine trail and the Mastodon Peak trail is 11.2 kilometers.
Page 12 Exercise 44b Answer
From the table, we know the Lost Palms Oasis trail is 11.6 km and the Skull Rock trail is 2.7 km. To find how much longer the Lost Palms Oasis trail is than the Skull Rock trail, we need to subtract 11.6 and 2.7.
To subtract two decimals, line up the decimals at their decimal places and then subtract each place value. Regroup if needed:
The Lost Palms Oasis trail is then 8.9 km longer than the Skull Rock Trail.
Result
8.9 km