6th Grade Math - Algebra Essentials

th
6

Grade Math - Algebra

By Jessica Corriere and Robert Richards
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Problem 1.What value of x makes the equation
below true?
7 + x = 84
A.
B.
C.
D.

12
77
83
91

How To Solve
There are 2 basic rules to a problem like this.
Isolate the unknown on one side of the equal sign.
Whatever action you perform must be performed on
both sides of the equal sign.
7 + x = 84
Subtract 7 from both sides to isolate x
7 + x – 7 = 84 – 7
x = 77
The correct answer is B, 77

Solve simple proportions within context
Problem 2. Guafi selects 15 pieces of fruit. If 3
out of 5 of these fruit are bananas, b, how many
bananas does Guafi select?
A. 3
B. 9
C. 13
D. 25

How To Solve
Facts provided;
There are 15 total pieces of fruit
There are 3 bananas for every 5 pieces of fruit
Write a formula that expresses those two statements;
15 pieces of fruit (3 out of every 5 pieces of fruit) is a banana.
G-Zibo Tip – In a word problem, the word “is” means equals in the formula
15 (3/5) = b
Whatever you do to one side of the problem you must do to the other side of the problem.
15 (3/5) = b
Simplify the left side
Multiply 15 times 3/5

Then you have

9=b
B is the correct answer

Translate two-step verbal sentences into
algebraic equations
Problem 3.What algebraic equation represents
“three times the difference of a number, x, and
nine equals fifteen”?
A.
B.
C.
D.

9x -3 = 15
3(9) – x = 15
9 – 3x = 15
3(x – 9) = 15

How To Solve
The statement gives 3 times the difference of x
and 9
3(x – 9) equals 15
The word equals determines where the equal
sign goes,
15 on one side,
Everything else on the other
3(x -9) = 15
Choice D is the correct answer

Evaluate formulas for given input values (circumference,
area, volume, distance, temperature, interest, etc.
Problem 4. A circle has a radius of 18 inches. What is the
circumference of the circle in terms of π?
G-Zibo Tip- C = 2πr
C = circumference
r = radius
π = pi = 3.14

A.
B.
C.
D.

36π
20π
18π
9π

How To Solve
The problem states “in terms of π”, that means
that π must be part of the answer.
Substitute 18 for the radius (r) in the formula
provided in the tip
C = 2 x π x 18
C (circumference) = 36π
The correct answer is A

Translate two-step verbal sentences into algebraic
equations
Problem 5.The number of solar panels, s, installed along
Oak Street is 3 less than 4 times the number of solar
panels, installed along Poplar Street. Which expression
can be used to determine the number of solar panels on
Oak Street?

A.
B.
C.
D.

= 7s
= 12s
= 3-4s
= 4s-3

How To Solve
Analyzing the statement it shows
Oak Street is (equals) 3 less (negative) than 4 times (multiply) Poplar street solar
panels, s,
In this case, 3 is a negative number because of the statement, "3 less than". 4 is a
positive number and then it is multiplied by the variable s because of the statement,
"4 times Poplar street solar panels, s".
= -3 + 4s
Now rearrange the formula to look like one of the answers above; Choice D is the
correct answer.
G-Zibo Tip: Many times word problems such as these can be very frustrating and
confusing. It is important that if you are feeling frustrated by this type of problem try
again after when you are feeling refreshed and have a clear mind. As the concept
sinks in by repeating the problem (practice, practice, practice) try substituting the
solar panels for another object and the street names for another location. Make up
your own word problem and solve (practice, practice, practice).

Solve and explain simple one step equations
using inverse operations involving whole
numbers
Problem 6.What is the value of n in the equation
below?
66 + n = 226
Show your work
Answer_________________

How To Solve
Whatever you do to one side of the problem you
must do to the other side of the problem.
66 + n = 226
Subtract 66 from each side
66 + n - 66 = 226 – 66
n = 160

Solve and explain simple one step equations using inverse
operations involving whole numbers
Problem 7. At a recycling center there are 16 pallets of
used car batteries to recycle. The manager places 8
batteries per pallet for shipping to the factory. He uses
the equation below to calculate the total number of
batteries, b, to recycle.
b/8 = 16
What is the total number of batteries to ship to the
factory?
Show your work
Answer_________________

How To Solve
Whatever action you perform must be performed
on both sides of the equal sign.
b/8 = 16
Multiply both sides by 8 to cancel out the 8 in b/8
b/8 x 8 = 16 x 8
b = 128

Solve and explain simple one-step equations using
inverse operations involving whole numbers
Problem 8. The tide on the island of Tonga changes
from 4.7 feet at high tide to 1.3 feet at low tide.
Write an equation that shows the change in tide
when x represents the number of feet the tide
increased.
Equation_________________
How many feet did the tide increase?
Show your work
Answer_________________

How To Solve
Set up a formula that will look for the difference
between the high tide and low tide.
High tide equaled 4.7 feet
Low tide equaled 1.3 feet
The equation would be
4.7 feet – 1.3 feet = x
3.4 feet = x
The tide increased by 3.4 feet

Evaluate formulas for given input values (circumference, area,
volume, distance, temperature, interest, etc.)
Problem 9. Rio measures one side of a solar panel. The length
is 8 feet and the width is one half of the length. What is the
area, in square feet, of the solar panel?
G-Zibo Tip, A = lw
A = Area
l = length
w = width

A.
B.
C.
D.

4
16
32
108

How To Solve
The length is 8 feet
The width is given as ½ the length (8 feet) = 1/2
x8=4
A = lw
Substitute the numbers for length and width;
A = 8 x ½(8)
A=8x4
A = 32
C is correct

Translate simple verbal expressions into algebraic
expressions
Problem 10. Demi brought 8 bottles of water to a picnic.
Of those bottles of water, she and her friends opened a
certain number of bottles, b. What expression can be
used to determine the number of unopened bottles at
the picnic?

A.
B.
C.
D.

8+b
b+8
8–b
b-8

How To Solve
Demi started with 8 bottles and she and her friends drank
some of the water. The number of bottles that were
drunk is then subtracted from the total.

Original total = 8 bottles
Bottles she and her friends opened = b
The number of bottles they drank is then subtracted from
the original number.
Choice C states that 8 – b is the answer, that is the correct
choice.

Solve simple one-step equations using basic wholenumber facts
Problem 11. Kennet brought 6 bags of cans to the
recycling center. He received a total of $12. He used the
equation below to determine the price per bag, b, which
he received.
6b = $12.00
How much money did Kennet receive for each bag?
A. $72.00
B. $18.00
C. $6.00
D. $2.00

How To Solve
Isolate the unknown on one side of the equal
sign. Then whatever action you perform must be
performed on both sides of the equal sign.
6b = $12.00
Divide both sides by 6
6b ÷ 6 = $12.00 ÷ 6
b = $2.00
Answer D is the correct choice.

Substitute assigned values into variable expressions
and evaluate using order of operations
Problem 12. What is the value of the expression
below when r = 2?
9 -3r =
A.
B.
C.
D.

0
3
6
12

How To Solve
The formula provided is
9 – 3r =
r was given a value of 2
r=2
Substitute 2 for r in the formula
9 – 3(2) =
Multiply the -3 times 2, which will equal -6
The problem now becomes
9–6=3
Choice B is the answer.

Translate simple verbal expressions into algebraic
expressions
Problem 12. An aquarium has 30 kinds of salt-water
fish. If f represents the number of tropical saltwater fish, which expression can be used to
determine the total number of fish that are not
tropical salt-water fish?
A. 30f
B. 30/f
B. 30 – f
D. 30 + f

How To Solve
The aquarium has 30 kinds of fish
f is the unknown number of tropical salt-water
fish
Whatever is not f would have to be the
remaining kinds of fish
30 – f =
Choice C has the correct formula.

Use substitution to evaluate algebraic
expressions (may include exponents of one, two
and three)
Problem 13. What is the value of the expression
6m + 33 when m equals 7?
Show your work

How To Solve
The formula given
6m + 33
The value of m is 7. Substitute 7 for m in the formula
6(7) + 33
Solve the exponent first. 33 = 3 x 3 x 3 = 27
6(7) + 27
Solve the multiplication of 6 times 7
42 + 27
Add the result
69 is the answer

Solve and explain simple one-step equations using
inverse operations involving whole numbers
Problem 14. Urie is planting a garden. He has a total
of 320 seedlings to plant. He has planted 14 so far.
The equation below can be used to determine the
number of seedlings, s, Urie still needs to plant.
14 + s = 320
What is the number of seedlings that Urie still
needs to plant?
Show your work

How To Solve
Whatever action you perform must be performed
on both sides of the equal sign.
14 + s = 320
Subtract 14 from both sides
14 + s - 14 = 320 – 14
s = 306
Urie has to plant 306 more seedlings

Solve simple one-step equations using basic
whole-number facts
Problem 15. If 8n = 96, what value of n makes
the equation true?
A.
B.
C.
D.

12
88
104
768

How To Solve
Isolate the unknown on one side of the equal
sign.
Whatever action you perform must be
performed on both sides of the equal sign.
8n = 96
Divide both sides by 8 to isolate the n
8n ÷ 8 = 96 ÷ 8
n = 12

Substitute assigned values into variable expressions and evaluate using
order of operations
Problem 16. Gabija has been growing 5 containers of avocado seeds in
his windows at home. Once they started to sprout he added 3 more
avocado seeds. The expression below shows the total number of
avocado seeds in Gabija’s window when a represents the number of
avocado seeds in each container.
Let a represent the number of avocado seeds per container.
3 + (a x 5)
If each container holds 8 seeds, how many total avocado seeds are in
Gabijo’s windows
A. 18
B. 28
C. 43
D. 55

How To Solve
Whatever action you perform must be performed on both sides of the
equal sign.

3 + (a x 5) =
The number of avocado seeds per container is 8, represented as a
a=8
Substitute 8 for a in the formula
3 + (8 x 5) =
Perform the multiplication first
8 x 5 = 40
Then the addition
3 + 40 = 43
The answer is C

Use substitution to evaluate algebraic expressions (may
include exponents of one, two, and three)
Problem 17. Arlen wrote the expression below for 3 of his
classmates to calculate the age of his tortoise.

n2 x 7 - 4
If n represents the number of classmates, what is the age
of Arlen’s tortoise?
A.
B.
C.
D.

12
18
36
59

How To Solve
The formula given
n2 x 7 – 4
The number of classmates, n, is given as 3
Substitute the number 3 for the n in the formula
That gives
32 x 7 – 4
32 is the same as 3 x 3, which equals 9
The formula now reads as
9x7–4
Perform the multiplication operation first
63 – 4
Then the subtraction
63 – 4 = 59
Choice D is correct; the tortoise is 59 years old

Evaluate formulas for given input values (circumference, area, volume,
distance, temperature, interest, etc..
Problem 18. Geena walks at a rate, r, of 4 miles per hour. What is the
distance, d that she walks in the time, t, of 3 hours?

G-Zibo Tip, d = rt
d = distance
r = rate of speed
t = time
A.
B.
C.
D.

1 mile
7 miles
12 miles
43 miles

How To Solve
The formula for calculating distance is provided
d = rt
The rate of speed, r, was provided, 4 miles per hour
The time, t, was provided, 3 hours
Substitute the provided numbers for the letter symbols

Multiply the numbers; the “hours” cancel each other
leaving just the unit of miles
d = 12 miles
Choice D is the correct answer

Solve simple one-step equations using basic whole-number
facts
Problem 19. A sixth grade class is wants to raise money to
have an aquarium so they decide to have a bake sale. The
students earn $6 for each cake they sell. To determine how
many cakes they need to sell to reach their goal of $144, the
class uses the equation below where c represents the
unknown number of cakes the students need to sell.

6c = 144
Part A
What is the value of c in the equation?
Show your work
Answer_________

How To Solve
Whatever action you perform must be performed on both sides of the equal sign.
The formula provided
6c = 144
Divide both sides by 6 to isolate the c
6c ÷ 6 = 144 ÷ 6
On the left side 6 divided by 6 equals 1 leaving C alone on that side
And on the right side there is, 144 ÷ 6 = 24
c = 24
24 cakes must be sold to earn $144
Part B
The teacher wrote the equation shown below for her students to solve to find the
number of angel fish, a, they could buy to add to the fish tank in their classroom using
the $144, if each angel fish costs $9
144/a = 9

How To Solve
equal sign.
The formula provided
6c = 144
Divide both sides by 6 to isolate the c
6c ÷ 6 = 144 ÷ 6
On the left side 6 divided by 6 equals 1 leaving C alone on that side
And on the right side there is, 144 ÷ 6 = 24
c = 24
24 cakes must be sold to earn $144

Part B
The teacher wrote the equation shown below
for her students to solve to find the number of
angel fish, a, they could buy to add to the fish
tank in their classroom using the $144, if each
angel fish costs $9
144/a = 9

How To Solve
equal sign.
144/a = 9
a is the denominator on the left side
Multiply both sides by a to remove it from the denominator
144/a x a = 9a
The a’s on the left cancel each other out
144 = 9a
Divide both sides by 9 to isolate the a
144 ÷ 9 = 9a ÷ 9
The 9’s on the right of the equal sign cancel each other
16 = a
The class can buy 16 angelfish with their $144

Use substitution to evaluate algebraic expressions
(may include exponents of one, two, and three)
Problem 20. Jorja’s teacher asked the students to
evaluate the expression 43 + 2y, when y = 8.
Three of the students wrote their answer on the
board

Which student evaluated the expression correctly?
Show your work

How To Solve
The formula provided is
43 + 2y
The value of y is given as 8
Substitute the 8 for y in the formula
43 + 2(8)
Perform the action required by the exponent first
43 + 2(8) =
(4 x 4 x 4) + 2(8) =
64 + 2(8) =
Now perform the multiplication of 2 x 8
64 + 2(8) =
64 + 16 = 80
Govad had the correct answer with 80

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