Chapter 3 Numeric And Algebraic Expressions
Section 3.3: Write And Evaluate Numerical Expressions
Page 131 Exercise 1 Answer
The airline company charges $49 for each overwight bag and $75 for each oversized bag. To find the total amount of fees collected for that flight, first, claculate the products 49 x 50 and 75 x 6, Since there were 50 overweight bags and 6 oversized bags, and then find the sum of those products.
49 x 50 + 75 x 6 = 2450 + 450 = 2900
Result
The total amount of fees collected for that night is $2900.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 132 Exercise 1 Answer
The Order of Operations is:
Evaluate parentheses and brackets from inside out.
Evaluate any powers.
Multiply and divide in order from left to right.
Add or subtract in order from left to right.
Evaluating the given expression using the order of operations gives:
\(\frac{1}{8}\left[6^3+(48 \div 6)\right]-20\)= \(\frac{1}{8}\left[6^3+8\right]-20\) Evaluate inside the parentheses by dividing.
= \(\frac{1}{8}[{216}+{8}]-20\) Evaluate the power.
= \(\frac{1}{8}\)[224] – 20 Evaluate inside the brackets by adding.
= 28 – 20 Multiply.
= 8 Subtract.
It is important to follow the order of operations because evaluating in a different order will not give the correct value.
Consider the expression 4 + 2 × 3. Order of operations states we need to multiply and then add so the correct value is 4 + 2 × 3 = 4 + 6 = 10. If we added and then multiplied we would get the incorrect value of 4 + 2 × 3 = 6 × 3 = 18.
Result
\(\frac{1}{8}\left[6^3+(48 \div 6)\right]-20\)= \(\frac{1}{8}\left[6^3+8\right]-20\) Evaluate inside the parentheses by dividing.
= \(\frac{1}{8}[{216}+{8}]-20\) Evaluate the power.
= \(\frac{1}{8}\)[224] – 20 Evaluate inside the brackets by adding.
= 28 – 20 Multiply.
= 8 Subtract.
It is important to follow the order of operations because evaluating in a different order will not give the correct value.
Page 133 Exercise 2a Answer
Find the product inside the parentheses.
\(3.2^2-[(9 \times 4)+9] \times\left(\frac{1}{3}\right)^2=3.2^2-[(36)+9] \times\left(\frac{1}{3}\right)^2\)Find the sum inside the brackets.
\(=3.2^2-[45] \times\left(\frac{1}{3}\right)^2\)Evaluate the power.
\(=10.24-45 \times\left(\frac{1}{3}\right)^2\)Evaluate the power.
\(=10.24-45 \times \frac{1}{9}\)Find the product.
= 10.24 – 5
Find the difference.
= 5.24
Result
5.24
Page 133 Exercise 2b Answer
To insert grouping symbols so that the expression has a value of 80 use the trial and error method. Try putting parentheses in some place and calculating the value. If the result isn’t what you need, try putting the parentheses in some other place. Repeat until you find the correct result.
For example:
\((6+12) \times\left(\frac{2}{3}\right)^2 \times 3+7\)
= \(18 \times\left(\frac{2}{3}\right)^2 \times 3+7\)
= 18 x \(\frac{4}{9}\) x 3 + 7
= 8 × 3 + 7
= 24 + 7
= 32
≠ 80
A second try:
= \(18 \times\left(\frac{2}{3}\right)^2 \times 10\)
= 18 x \(\frac{4}{9}\)
= 8 × 10
= 80
Result
\((6+12) \times\left(\frac{2}{3}\right)^2 \times(3+7)\)Page 134 Exercise 1 Answer
To write a numerical expression, we use the given relationships in the problem to determine what operations are needed in the expression. To evaluate the expression, we then use the order of operations.
Page 134 Exercise 2 Answer
From the Order of Operations, we know that evaluating inside of parentheses and brackets must be done first. Grouping symbols can then change the value of a numerical expression because they change what part of the expression is evaluated first.
We are given the expression
80 ÷ 8 x 5 + 42.
Grouping the first two numbers and then evaluating gives:
(80 ÷ 8) x 5 + 42
= 10 x 5 + 42 Evaluate inside the parentheses by dividing.
= 10 x 5 + 16 Evaluate the power.
= 50 + 16 Multiply.
= 66 Add.
Grouping the second and third numbers and then evaluating gives:
80 ÷ (8 x 5) + 42
= 80 ÷ 40 + 42 Evaluate inside the parentheses by multiplying.
= 80 ÷ 40 + 16 Evaluate the power.
= 2 + 16 Divide.
= 18 Add.
Grouping the last two numbers and then evaluating gives:
80 ÷ 8 x (5 + 42)
= 80 ÷ 8 x (5 + 16) Evaluate inside the parentheses by evaluating the power.
= 80 ÷ 8 x 21 Evaluate inside the parentheses by adding.
= 10 x 21 Divide.
= 210 Multiply.
Grouping the last three numbers and then evaluating gives:
80 ÷ (8 x 5 + 42)
= 80 ÷ (8 x 5 + 16) Evaluate inside the parentheses by evaluating the power.
= 80 ÷ (40 + 16) Evaluate inside the parentheses by multiplying.
= 80 ÷ 56 Evaluate inside the parentheses by adding.
= \(\frac{10}{7}\) Divide.
Result
Grouping symbols can change the value of a numerical expression because they change what part of the expression is evaluated first.
(80 ÷ 8) × 5 + 42 = 66
80 ÷ (8 × 5) + 42 = 18
80 ÷ 8 × (5 + 42) = 210
80 ÷ (8 × 5 + 42) = \(\frac{10}{7}\)
Page 134 Exercise 3 Answer
First perform the operations in brackets, when that is done you are left with (7) × (9) ÷ (8). Since there are multiplication and division left, start from left to right.
Thus, the last operation you should perform is division.
Result
Division.
Page 134 Exercise 4 Answer
Charles says that 2 × 3 − 2 is 4, and Seth says that 2 × 3 − 2 is 2.
2 × 3 − 2 … First we multiply the numbers from left to right.
= 6 − 2 … Than subtract the numbers.
= 4
The correct answer is 4, so Charles is correct.
How did Seth find the answer 2 and why is it wrong?
2 × 3 − 2 = 2 × 1 = 2
Seth first subtracted 2 from 3 and than multiplied the result by 2. However, we know that when there is no parenetheses nor brackets we have a specific order of operations. First, evaluate the parentheses and brackets from inside out. Second, evaluate powers. Third, multiply and divide from left to right. Fourth, Add and subtract from left to right. Seth was wrong to have first subtracted 2 from 3 and than multipled the result by 2.
Result
Charles is correct.
Page 134 Exercise 5 Answer
52 + (6.7−3.1) … Evaluate the expression inside the parentheses.
= 52 + 3.6 … Evaluate the power.
= 25 + 3.6 … Add the numbers.
= 28.6
Result
28.6
Page 134 Exercise 6 Answer
(8.2 + 5.3) ÷ 5 … Evalute the expression inside the parentheses.
= 13.5 ÷ 5 … Divide the numbers.
= 2.7
Result
2.7
Page 134 Exercise 7 Answer
(1.5−0.52) ÷ [(3 + 2) × 2] … Add the numbers inside the parentheses.
= (1.5 − 0.52)÷[5 × 2] … Evalute the expression inside the brackets.
= (1.5 − 0.52) ÷ 10 … Evaluate the power.
= (1.5 − 0.25) ÷ 10 … Evaluate the expression inside the parentheses.
= 1.25 ÷ 10 … Divide the numbers.
= 0.125
Result
0.125
Page 134 Exercise 8 Answer
36.8 ÷ [11.5 − (2.5×3)]2 … Evalute the expression inside the parentheses.
= 36.8 ÷ [11.5 − 7.5]2 … Evalute the expression inside the brackets.
= 36.8 ÷ 42 … Evalute the power.
= 36.8 ÷ 16 … Divide the numbers.
= 2.3
Result
2.3
Page 134 Exercise 9 Answer
The Order of Operations is:
Evaluate parentheses and brackets from inside out.
Evaluate any powers.
Multiply and divide in order from left to right.
Add or subtract in order from left to right.
Using the order of operations then gives:
6 + 4 × 5 ÷ 2 − 8 × 1.5
= 6 + 20 ÷ 2 – 12 Multiply.
= 6 + 10 – 12 Divide.
= 16 – 12 Add.
= 4 Subtract.
Result
4
6 ÷ 4 × 5 ÷ 2 − 8 × 1.5 … Multiply and divide from left to right.
= 1.5 × 5 ÷ 2 − 8 × 1.5 … Multiply and divide from left to right.
= 7.5 ÷ 2 − 8 × 1.5 … Multiply and divide from left to right.
= 3.75 − 8 × 1.5 … Multiply and divide from left to right.
= 3.75 − 12 … Add and subtract from left to right.
= −8.25
Result
−8.25
Page 134 Exercise 10 Answer
The given expression is 12 × 32 + 36 and the target value is 540.
12 × 32 + 36 = 12 × 9 + 36 = 108 + 36 = 144 ≠ 540
12 × (32 + 36) = 12 × (9 + 36) = 12 × 45 = 540
Result
12 × (32 + 36)
Page 134 Exercise 11 Answer
The given expression is 32 ÷ 23 − 4 and the target value is 8.
32 ÷ 23 − 4 = 32 ÷ 8 − 4 = 4 − 4 = 0 ≠ 8
32 ÷ (23 − 4) = 32 ÷ (8 − 4) = 32 ÷ 4 = 8
Result
32 ÷ (23 − 4)
Page 134 Exercise 12 Answer
The given expression is 2.32 + 9 × 4 ÷ 2 and the target value is 28.58.
2.32 + 9 × 4 ÷ 2 = 5.29 + 9 × 4 ÷ 2 = 5.29 + 45 ÷ 2 = 5.29 + 22.5 = 27.79 ≠ 28.58
(2.32 + 9) × 4 ÷ 2 = (5.29 + 9) × 4 ÷ 2 = 14.29 × 4 ÷ 2 = 57.16 ÷ 2 = 28.58.
Result
(2.32 + 9) × 4 ÷ 2
Page 135 Exercise 13 Answer
42 − (3.1 + 6.4) + 4.5 … Evaluate the expression in the parentheses.
= 42 − 9.5 + 4.5 … Evaluate the power.
=16 − 9.5 + 4.5 … Add and subtract numbers from left to right.
= 6.5 + 4.5 … Add and subtract numbers from left to right.
= 11
Result
11
Page 135 Exercise 14 Answer
\((8.7+3.3) \times\left(\frac{1}{2}\right)^2\) …..Evaluate the expression in the parentheses.
= \(12 \times\left(\frac{1}{2}\right)^2\) …….Evaluate the power.
= 12 x \(\frac{1}{4}\) …..Multiply the numbers.
= 3
Result
3
Page 135 Exercise 15 Answer
157.8 − (32 + 6) × 3 … Evaluate the power.
= 157.8 − (9 + 6) × 3 … Evaluate the expression in the parentheses.
= 157.8 − 15 × 3 … Multiply the numbers from left to right.
= 157.8 − 45 … Subtract the numbers from left to right.
= 112.8
Result
112.8
Page 135 Exercise 16 Answer
4.3 + (8.4 − 5.1) … Evalute the expression in the parentheses.
= 4.3 + 3.3 … Add the numbers.
= 7.6
Result
7.6
Page 135 Exercise 17 Answer
\(1.25 \times 4+3 \times 2 \div\left(\frac{1}{2}\right)^3\) …..Evaluate the power.
= \(1.25 \times 4+3 \times 2 \div \frac{1}{8}\) …. Multiply and divide the number from left to right.
= \(5+3 \times 2 \div \frac{1}{8}\) ….Multiply and divide the number from left to right.
= 5 + 6 ÷ \(\frac{1}{8}\) …Multiply and divide the number from left to right.
= 5 + 6 x \(\frac{8}{1}\) … Multiply and divide the number from left to right.
= 5 + 48 …. Add the numbers.
= 53
Result
53
Page 135 Exercise 18 Answer
[23 x (152 ÷ 8)] − 52 … Evalute the expression in the parentheses.
= [8 × 19] − 52 … Evalute the power.
= 152 – 52 … Subtract the numbers.
= 100
Result
100
Page 135 Exercise 19 Answer
The given expression is 2 × 9 + 7 and the target value is 32.
(2 × 9) + 7 = 19 + 7 = 26 ≠ 32
2 × (9 + 7) = 2 × 16 = 32
Result
2 × (9 + 7)
Page 135 Exercise 20 Answer
The given expression is \(\frac{1}{3}\) x 21 – 3 and the target value is 6.
\(\frac{1}{3}\) x 21 – 3 = 7 – 3 = 4
\(\frac{1}{3}\) x (21 – 3) = \(\frac{1}{3}\) x 18 = 6
Result
\(\frac{1}{3}\) x (21 – 3)
Page 135 Exercise 21 Answer
The given expression is 2.5 + 5 × 6 − 2 and the target value is 43.
2.5 + 5 × 6 − 2 = 2.5 + 30 − 2 = 30.5 ≠ 43
(2.5 + 5) × 6 − 2 = 7.5 × 6 − 2 = 45 − 2 = 43
Result
(2.5 + 5) × 6 − 2
Page 135 Exercise 22 Answer
Cory bought some baseball equipment, a bat, a glove, and 3 baseballs. A bat costs $69, a glove $75, and a baseball $5.50. He used a coupon for \(\frac{1}{2}\) off the price of the bat and glove. \(\frac{1}{2}\) × (69 + 75) + 3 × 5.50 … Evalute the expression in the parentheses. = \(\frac{1}{2}\) × 144 + 3 × 5.50 … Multiply the numbers from left to right. = 72 + 16.50 … Add the numbers. = 88.50$
Result
The total cost of the bat, the glove, and 3 baseballs is $88.50.
Page 135 Exercise 23 Answer
First, calculate the value of the given expression.
5 + (8 − 4) ÷ 2 + 3
Find the difference inside the parenthesis.
= 5 + (4) ÷ 2 + 3
Find the quotient.
= 5 + 2 + 3
Find the sum.
= 7 + 3
Find the sum.
= 10
To write a numerical expression with the same value as the previous one, 10, start with that value. Rewrite it as a result of one of the arithmetic operations.
For example, write 10 as a difference of two numbers.
10 = 20 − 10
Rewrite one of the numbers as a product of two of its factors.
10 = 20 − 10 = 4 × 5 − 10
Rewrite one of the numbers as a power.
10 = 20 − 10 = 4 × 5 − 10 = 22 × 5 − 10
A numerical expression with the same value as the one given in the Exercise:
22 × 5 − 10
For example, write 10 as a product of two of its factors.
10 = 2 × 5
Rewrite one of the factors as a quotient.
10 = 2 × 5 = \(\frac{8}{4}\) × 5
Rewrite one of the numbers as a difference.
10 = 2 × 5 = \(\frac{8}{4}\) × (10 − 5)
A numerical expression with the same value as the one given in the Exercise: \(\frac{8}{4}\) × (10−5).
Result
Possible answers: 22 × 5 − 10 and \(\frac{8}{4}\) × (10−5)
Page 136 Exercise 24 Answer
The first part of the numerical expression to evaluate is the part inside the parentheses or bracket. However, in this example, there are more than one parentheses, so the first part we evaluate is the power since powers come second in the order of operations.
(26 + 2.5) − [(8.3 × 3) + (13 − 0.25)]
=(26 + 2.5) − [(8.3×3) + (1−0.25)]
Next, we evaluate the product.
= (26 + 2.5) − [(8.3 × 3)+(1 − 0.25)]
= (26 + 2.5) − [(24.9)+(1−0.25)]
Find the sum.
= (26 + 2.5) − [(24.9)+(1−0.25)]
= (28.5) − [(24.9)+(1−0.25)]
Find the difference.
= (28.5) − [(24.9)+(1−0.25)]
= (28.5) − [(24.9)+(0.75)]
Find the sum inside the brackets.
= (28.5) − [(24.9)+(0.75)]
= (28.5) − [25.65]
Find the difference.
= (28.5) − [25.65]
= 2.85
Result
The first part of the numerical expression to evaluate is the part inside the parentheses or bracket. However, in this example, there are more than one parentheses, so the first part we evaluate is the power since powers come second in the order of operations.
Page 136 Exercise 25 Answer
Solve the numerical expression and then compare it to Evan’s solution.
0.22 + 12 ÷ (1.5 × 4)
Evalute the power.
= 0.04 + 12 ÷ (1.5 × 4)
Find the product inside the parentheses.
= 0.04 + 12 ÷ (6)
Find the quotient.
= 0.04 + 2
Find the sum.
= 2.04
Evan’s solution is not correct. The solution is 2.04.
Result
Evan’s solution is not correct. The solution is 2.04.
Page 136 Exercise 26 Answer
Since the length of the rectangular drawing is 12 inches, one-third of that length is the product \(\frac{1}{3}\) × 12. Add three to that product.
\(\frac{1}{3}\) × 12 + 3
Find the product.
= 4 + 3
Find the sum.
= 7
The width of the rectangular drawing is 7 inches.
To find the perimeter calculate the sum of all sides of the rectangular.
12 + 7 + 12 + 7 = 38
Result
The perimeter of the drawing is 38 square inches.
Page 136 Exercise 27 Answer
Even though the expressions have the same numbers and the same operations, they are not the same since Frederick’s expression has parentheses and brackets in multiple places and Lana’s doesn’t. Frederick has then carried out the operations in a different order than Lana, and thus the results are different but both correct.
[(53.7+37.2) − (33 + 3.8)] − 8.6 = 51.5
Frederick first evaluated the power.
= [(53.7+37.2) − (27+3.8)] − 8.6
He than found the sum in the parentheses.
= [(53.7 + 37.2) − (30.8)] − 8.6
He found the sum in the other parentheses.
= [(90.9) − (30.8)] − 8.6
He found the difference in the brackets.
= [60.1] − 8.6
Finally, he found the difference, the end result.
= 51.5
53.7 + 37.2 − 33 + 3.8 − 8.6 = 59.1
Lana also first evaluated the power.
= 53.7 + 37.2 − 27 + 3.8 − 8.6
However, in her expression she doesn’t have any parentheses or brackets, so next she found the sums and differences – from left to right.
= 90.9 − 27 + 3.8 − 8.6
= 63.9 + 3.8 − 8.6
= 67.7 − 8.6
= 59.1
Result
Even though the expressions have the same numbers and the same operations, they are not the same since Frederick’s expression has parentheses and brackets in multiple places and Lana’s doesn’t. Frederick has then carried out the operations in a different order than Lana, and thus the results are different but both correct.
Page 136 Exercise 28 Answer
Lillian both four hairbrushes at $3.99 each. She had a coupon for $1 off. Her mom paid for half of the remaining cost.
\(\frac{1}{2}\) × [(4×3.99) − 1] … Evalute the expression in the parentheses.
= \(\frac{1}{2}\) × [15.96−1] … Evalute the expression in the brackets.
= \(\frac{1}{2}\) × 14.96 … Multiply the numbers.
= 7.48
Result
Lillian paid $7.48 toward the purchase of the hairbrushes.
Page 136 Exercise 29 Answer
In an ecosystem, some animals get energy by eating plants. An elk can eat 20 pounds of plants each day and we need to find how many pounds of plants a herd of 18 elk can eat in one week.
20 × 18 × 7 … Multiply the numbers from left to right.
= 360 × 7 = 2520
Result
A herd of 18 elk can in one week eat 2520 pounds of plants.
Page 136 Exercise 30 Answer
12.3 × [(2×1.7) + 6.6]
Find the product in the parentheses.
= 12.3 × [(3.4) + 6.6]
Find the sum in the brackets.
= 12.3 × [10]
Find the product.
= 123
24 ÷ [(3.2 × 0.8) + 1.44]
Find the product in the parentheses.
= 24 ÷ [(2.56) + 1.44]
Find the sum in the brackets.
= 24 ÷ [4]
Evaluate the power.
= 16 ÷ [4]
Find the quotient.
= 4
6.2 + (3 × \(\frac{1}{3}\) + 4.8)
Find the product in the parentheses..
= 6.2 + (1 + 4.8)
Find the sum in the parentheses.
= 6.2 + (5.8)
Find the sum.
= 12
[4 x (9.6 ÷ 3)] + 8.2
Find the quotient in the parenetheses.
= [4 x (3.2)] + 8.2
Find the product in the brackets.
= [12.8] + 8.2
Find the sum.
= 21
Result
12.3 × [(2 × 1.7) + 6.6] = 123
24 ÷ [(3.2 × 0.8) + 1.44] = 4
6.2 + (3 × \(\frac{1}{3}\) + 4.8) = 12
[4 × (9.6÷3)] + 8.2 = 21
Page 136 Exercise 31 Answer
[4 × (6.6÷3)] + 18 (Find the quotient in the parenthesis.)
= [4 × (2.2)] + 18 (Find the product in the brackets.)
= [8.8] + 18 (Find the sum.)
= 26.8
18.9 × [(2 × 2.7) − 4.6] − 22 (Find the product in the parenthesis.)
= 18.9 × [(5.4) − 4.6] − 22 (Find the difference in the brackets.)
= 18.9 × [0.8] − 22 (Evaluate the power. )
= 18.9 × [0.8] − 4 (Find the product. )
= 15.12 − 4 (Find the difference.)
= 11.12
33 ÷ [(2.6 x 0.7) + 1.18] (Find the product in the parenthesis.)
= 33 ÷ [(1.82) + 1.18] (Find the sum in the brackets.)
= 33 ÷ [3] (Evaluate the power. )
= 27 ÷ [3] (Find the quotient. )
= 9
6.9 + (2 x 42 – 4.1) (Evaluate the power inside the parenthesis)
= 6.9 + (32 – 4.1) (Find the product in the parenthesis)
= 6.9 + (32 – 4.1) (Find the difference in the parenthesis)
= 6.9 + (27.9) (Find the sum)
= 34.8
Result
[4 x (6.6 ÷ 3)] + 18 = 26.8
18.9 × [(2×2.7) − 4.6] − 22 = 11.12
33 ÷ (2.6 × 0.7) + 1.18] = 9
6.9 + (2 × 42 − 4.1) = 34.8