Expressions and Equations
Ratios and Proportions
Fractions and Mixed Numbers
Percent and Decimals
Miscellaneous
100
What is the value of x when 7x = 56?
x = 8
You are doing algebra here!
Divide both sides by 7 (opposite operation done with variable) You are really trying to isolate the variable. Get it by itself.
7x divided by 7 = x
56 divided by 7 = 8
so x=8
100
2 out of 4 students like chocolate ice cream. If there are 20 students, how many like chocolate ice cream?
10
Set up a ratio, then a proportion. Keep your parts with your parts and your whole with your wholes.
2 part/4 whole = ? part/ 20 whole
2/4 =?/20 How many times will 4 go into 20? 5 times. What you do to the denominator; you do to the numerator. SO, multiply 2 x 5 which gives you 10.
Are these equivalent? Use the butterfly method to check. 2 x 20 = 40 and 4 x 10 = 40. You are CORRECT!
100
5/8 ÷ 1/4
16 1/2
KEEP CHANGE FLIP
5/8 X 4/1 = 20/8 =
20 divided by 8
2 eights will go into 20 with 4 left over
2 4/8 Reduce the fraction.
Divide 4 and 8 by 2 = 1/2
Final answer: 2 1/2
OR draw fraction bars to solve.
100
25 % of 12?
3
Four ways to to do this...
1. Change the percent to a decimal and multiply
.25 x 12 = 3.00 = 3
2. Set up a proportion.
25% means 25 out of 100 ...25/100
25 (part)/100 (whole) = ? (part)/ 12 (whole)
25/100 = ? /12 Cross multiply 12 x 25 =300. Then divide by 100. 300 ÷ 100 =3
3. If you know that 25% = 25/100 = 1/4...
Then you want 1/4 of 12. OF means to multiply in this case. 1/4 x 12 = 1/4 x 12/1 = 12/4
12/4 = 12 ÷ 4 = 3
4. Draw a fraction bars.
100
Which statement is true?
A 0.04 < 0.004
B 8.2 < 8.02
C 7.6 < 7.67
D 0.03 < 0.003
Line up by the decimal to compare. (the point goes to the smallest number... 3< 4)
0.04
0.004 This number is less than the first. (Not correct)
8.2
8.02 This number is less than the first. (Not correct.)
7.6
7.67 (This number is greater than the first so it IS CORRECT.)
0.03
0.003 This number is less than the first number. NOT CORRECT.
200
(5r + 1/5 )2 when r = 0?
1/25
This is a substitution problem. So rewrite it first.
(5 (0) + 1/5)2 Work out what is inside the parenthesis first. Any number beside a variable indicates multiplication.
5 times 0 is 0 so
(0 + 1/5)2 Next, add what is inside.
0 + 1/5 = 1/5
Now, you have (1/5)2 = 1/5 x 1/5 = 1/25
200
There are 4 trucks for every 5 cars in a parking lot. How many trucks and cars could be in the parking lot?
A 64 trucks and 80 cars
B 72 trucks and 73 cars
C 84 trucks and 100 cars
D 96 trucks and 110 cars
A
4 goes into 64, 16 times and 5 goes into 80, 16 times (Both 64 and 80 are multiples of 16.)
200
Jamal will cut a piece of wood that is 5 1/2 feet long into 1 4 -foot sections. How many sections will result?
A 10
B 1 3/8
C 3/4
D 22
22
You are dividing the wood into equal parts...division.
5 1/2 divided by 1/4. Change mixed number to an improper fraction 11/2.
11/2 divided by 1/4, so...
KEEP CHANGE FLIP
11/2 X 4/1 = 44/2 = 22
200
What is the volume of a rectangular prism that has the following dimensions?
length 5 inches
width 4.2 inches
height 8.3 inches
Volume is length x width x height.
5 x 4.2 x 8.3 = 174.3 cubic inches
200
What is the value of (52 – 5) ÷ 5 × 23?
32
PEMDAS
Parenthesis first. 52 = 5 x 5 so 25-5 = 20
20 ÷ 5 x 23
Exponent
20 ÷ 5 x (2 x 2 x 2)
20 ÷ 5 x 8
Multiplication or Division (which ever comes first)
First, 20 ÷ 5 =4
Then 4 x 8 = 32
300
Which expression is equal to 48 + 72?
A 2(24 + 38)
B 8(6 + 12)
C 3(14 + 24)
D 24(2 + 3)
D
This is the distributive property. On the outside of the parenthesis, there must be a COMMON factor of the numbers given. When multiplied, each number should be equivalent to the original problem.
A. 2 x 24 = 48 but 2 x 38 = 76 NOT
B. 8 x 6 = 48 but 8 x 12 = 96 NOT
C. 3 x 14 = 42 and 3 x 24 = 72 NOT
D. 24 x2 = 48 and 24 x 3 = 72 YES!!!!
300
A restaurant orders corn tortillas and flour tortillas. The ratio of the number of corn tortillas to the number of flour tortillas is 2 : 3. What is the ratio of the number of flour tortillas to the total number of tortillas?
A 3 : 8
B 3 : 5
C 2 : 8
D 2 : 5
B
There are 3 flour tortillas and the total number of tortillas is 2 + 3 = 5. 3/5 = 3:5
**ALWAYS REREAD THE QUESTION TO MAKE SURE YOUR ANSWER MAKES SENSE.
300
Steve has 2 2/4 cups of flour. One of his recipes requires 1/2 cup of flour. What is the maximum number of times Steve can complete the recipe?
5
You are dividing flour into equal parts.
2 2/4 divided by 1/2
Change 2 2/4 into a mixed number.
10/4 divided by 1/2
KEEP CHANGE FLIP
10/4 X 2/1 = 20/4 = 5
300
Jenny bought gas for her car. • Gas cost $3.45 per gallon. • Jenny bought 12.2 gallons. What was the total cost for Jenny’s gas?
42.09
(Notice you do not include the $ sign when you have to put in the number.)
You are multiplying decimals here. JUST DO IT! (multiply as usual) You will have to move (SWOOSH) the decimal three places because there are three numbers behind the decimal.
3.45 X 12.2 = 42090. Move the decimal from right to left 3 times. (3 swooshes) 42.090
It is money so you will drop the last 0. = 42.09
300
What is the surface area of the net?
170 cm2
You must know how to get the areas of a triangle and a rectangle.
There are three rectangles the same size. So length times width is 10 x 5 = 50 For each 3 is 50 x 3 =150 cm2 (rectangles area)
Now, the triangles
base x height x 1/2 OR base x height ÷ 2
5 x 4 x 1/2 = 20/2 =10 There are two triangles the same size so 10 x 2 = 20 cm2.
ADD ALL AREAS
150 + 20 = 170 cm2
400
What is the value of 1,608 ÷ 103?
1.608
You can look at this two ways.
1. 103 = 10 x 10 x10 =1000 so 1,608 ÷ 1,000 is 1.608
2. Or when dividing whole numbers by powers of 10, you can move the decimal to the left. In this case it would move three times as indicated by the exponent. 1608. (Decimal is at the end of a whole number.) Jump three places between the numbers to the left. You will land between the 1 and 6. So the answer is 1.608
****Think...the number should get smaller.
400
A set of stickers contains 4 hearts for every 6 stars. Which choice contains an equivalent ratio of hearts to stars?
A 6 hearts to 9 stars
B 2 hearts to 4 stars
C 1 heart to 3 stars
D 8 hearts to 10 stars
A
4/6 =2/3 so does 6/9 = 2/3? SURE! (Sometimes you have to simplify to see what makes sense.) ****Here is where you CAN use cross multiplication to see what fractions are equal. Hint: Set them up as equivalent fractions and then cross multiply. IF they are the same, they are equal.
400
What is the value of x when 1/3 x = 3 2/3 ?
11
To find x, you have to do the opposite operation (inverse) so divide...
Divide both sides by 1/3.
1/3 divided by 1/3 is 1 so that leaves 1x = x
3 2/3 divided by 1/3
Change mixed number to improper fraction.
11/3 divided by 1/3
KEEP CHANGE FLIP
11/3 x 3/1 = 33/3 = 11
400
Sarah earns $400 per week and spends 15% of her earnings on transportation. How much does Sarah spend on transportation every week?
You are finding percent of. There are four ways. See slide one.
Here is one way.
15% of $400 Change percent to a decimal. Multiply.
400 x .15 = 60.00
400
Find the surface area.
54 in2
This is a cube net, made of all squares. All the squares are the same size in a cube.
Area of a square is l x w or
S2 (side squared or s x s).
One square is 3 x 3 = 9 in2
There are six squares so 6 x 9 = 54 in2
500
Tonya pays $300 each month to rent an office where she earns $25 per hour tutoring students. Which equation represents Tonya’s profit, y, for working x hours?
A y = 25 + 300x
B y = 25x - 300
C y = 25 – 300x
D y = 25x + 300
B
Y is what Tonya will have left over after she makes money and then pays her bill.
Y= 25x (25 is what she makes but we do not know how many hours which is the variable) She pays her bill with this money. You multiply the money times the hours to get her pay.
300 is her bill or money owed which means it has to be subtracted from the money she makes above.
Therefore, Y (money left) = 25x (money made) - 300 (money owed)
500
Which is the better buy?
15 ounces $1.29 OR 18 ounces $2.26
How do you know?
15 ounces for $1.29 because to find unit rate you divide the cost by the unit (You will also need to know how to compare decimals for this or round money.)
1.29 ÷ 15 is 0.086 which would be rounded to 9 cents
$2.26 ÷ 18 is 0.12555... or rounded to 0.13, 13 cents
9 cents per ounce is a better buy
500
Sam is using the expression m/8n in class. What is the value of the expression when m = 4 and n = 1/2 ?
1
This is a substitution problem so you REWRITE it.
If m=4, 4/8n
If n= 1/2 4/8(1/2)
That means 4 ÷ 8(1/2) **a number beside a parenthesis means to multiply 8 x 1/2
Work out your numerator. Multiply 8 x 1/2 you get 8/1 X 1/2 (Put whole numbers over 1.)
8/1 X 1/2 = 8/2 = 4
You NOW have 4/4. Any number over itself is 1.
500
At a middle school, 74 students have freckles. There are 258 students in the school. To the nearest tenth of a percent, what percent of the students have freckles?
A 18.4%
B 28.7%
C 34.6%
D 40.2%
28.7%
You have to set up a ratio. 74 out of 258
74/258 The line means divide.
74 ÷ 258 = 0.2868..move the decimal to the right two times to create percent.
28.68 THEN ROUND to closest TENTH. That is where the 6 is. 8 is greater than 4 so you round UP one.
28.7%
500
What is the area of the triangle?
15 u2
Find the base. (This is what the triangle sits on.) Count from how many lines from point to point. 6 units
Find the height (How tall is it?) Count the lines from the base to the point. There are 5 units.
Multiply 6 x 5 (base x height) = 30 square units
Then, divide base x height by 2. 30 ÷ 2 = 15 u2