Math Placement Practice - Stage 1: Refresh and Learn

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Math Placement Practice - Stage 1: Refresh and Learn

How to Use This Guide

Use as much or as little as you need.
Familiarize yourself with the process to solve the problems.
Write out the steps — it helps you absorb the material.
Use the linked YouTube videos (created by us) as a mini-lesson.

Arithmetic

Multiplication and Division of Integers

Example 1: $436 \times 52$

You need to multiply 436 and 52. In this example 1 video, the traditional multiplication algorithm is demonstrated — but use any method you are comfortable with.

Example 2: $624 \div 16$

You need to divide 624 by 16. In this example 2 video, long division is demonstrated. Again, use any method that you are comfortable with.

Fractions

Example 3: ${(\frac{5}{6})}^{2}$

Example 3 video. An exponent is shorthand for repeated multiplication. Since the exponent is 2, multiply the fraction by itself twice: $(\frac{5}{6}) (\frac{5}{6})$ . When multiplying fractions, multiply straight across — numerators with numerators and denominators with denominators.

{(\frac{5}{6})}^{2} = (\frac{5}{6}) (\frac{5}{6}) = \frac{5 (5)}{6 (6)} = \frac{25}{36}

Since 25 and 36 share no common factors, the fraction is in simplest form.

Example 4: $\frac{7}{8} \div \frac{1}{3}$

Example 4 video. As a general strategy, we do not perform division of fractions directly. Instead, multiply the first fraction by the reciprocal of the second — “keep, change, flip.” Keep the first fraction, change division to multiplication, and flip the second fraction.

\frac{7}{8} \div \frac{1}{3} = \frac{7}{8} \cdot \frac{3}{1} = \frac{7 (3)}{8 (1)} = \frac{21}{8}

Since 21 and 8 share no common factors, this fraction is in simplest form.

Example 5: $\frac{2}{9} + \frac{3}{7}$

Example 5 video. The fractions $\frac{2}{9}$ and $\frac{3}{7}$ do not share a denominator, so we find equivalent fractions with matching denominators. Multiply ninths by $\frac{7}{7}$ and sevenths by $\frac{9}{9}$ so both denominators become 63: $\frac{2}{9} \cdot \frac{7}{7} = \frac{14}{63}$ and $\frac{3}{7} \cdot \frac{9}{9} = \frac{27}{63}$ . With a common denominator, add the numerators over the common denominator.

\frac{14}{63} + \frac{27}{63} = \frac{41}{63}

Since 41 and 63 share no common factors, this fraction is in simplest form.

Example 6: Write 0.6 as a fraction in simplest terms

Example 6 video. Remember place value. 0.6 is read as “six tenths,” which is $\frac{6}{10}$ . Since 6 and 10 share a common factor of 2, this is not in simplest terms. Dividing out the common factor gives $\frac{3}{5}$ .

Similarly, to write 0.35 as a fraction in simplest terms: 0.35 is “thirty-five hundredths,” written $\frac{35}{100}$ . Both share a factor of 5, so it simplifies to $\frac{7}{20}$ .

Absolute Value

Example 7: $| - 15 + 8 |$

Example 7 video. Any operation inside absolute-value bars must be computed first. Compute $- 15 + 8 = - 7$ , then take $| - 7 | = 7$ .

| - 15 + 8 | = | - 7 | = 7

Prealgebra and Algebra

Evaluate (Order of Operations)

Order of operations video. The order of operations exists so we arrive at a unique answer. Other orderings can give different results, so it’s important to follow the order correctly.

Recall the order of operations: PEMDAS

Parentheses: Compute any operation in parentheses Exponents: Evaluate powers and roots Multiplication and Division: Whichever comes first, left to right Addition and Subtraction: Whichever comes first, left to right

Example 8: $- 16 \div 2 (1 + 3)$

Add inside the parentheses first: $- 16 \div 2 (4)$ .
Handle multiplication and division left to right. Division comes first: $- 16 \div 2 = - 8$ .
Multiply: $- 8 (4) = - 32$ .

Simplify the Expression

Example 9: $7 x - 3 + 8 x + 9$

Example 9 video. There are two kinds of terms:

Terms with $x$ : $7 x + 8 x = 15 x$ .
Constant terms: $- 3 + 9 = 6$ .

7 x - 3 + 8 x + 9 = 15 x + 6

In the form $a x + b$ : $a = 15$ , $b = 6$ .

Example 10: $- x^{2} + 7 x - 2 + 5 x^{2} + 3 x - 6$

Example 10 video. There are three 'types' of terms here:

Terms with $x^{2}$ : $- x^{2} + 5 x^{2} = 4 x^{2}$ .
Terms with $x$ : $7 x + 3 x = 10 x$ .
Constants: $- 2 - 6 = - 8$ .

In the form $a x^{2} + b x + c$ : $4 x^{2} + 10 x - 8$ , so $a = 4$ , $b = 10$ , $c = - 8$ .

Substitution

Example 11: Evaluate $- 7 x + 5$ when $x = - 3$

Example 11 video. Replace $x$ with $- 3$ , then simplify using the order of operations.

- 7 (- 3) + 5 = 21 + 5 = 26

Solving Equations

The goal is to isolate the variable. Undo the order of operations one step at a time using opposite operations.

Example 12: Solve $- 5 x = 30$

Example 12 video. Since $x$ is multiplied by $- 5$ , divide both sides by $- 5$ to isolate $x$ : $x = - 6$ . Check: $- 5 (- 6) = 30$ . ✓

Example 13: Solve $x - 3 = 17$

Example 13 video. Add 3 to both sides: $x = 20$ . Check: $20 - 3 = 17$ . ✓

Examples 12 and 13 are one-step equations. Combinations of these steps solve multi-step equations. Work from the terms farthest from the variable inward, moving $x$ terms to one side and constants to the other.

Example 14: Solve $4 x + 6 = 16$

Subtract 6 from both sides: $4 x = 10$ .
Divide both sides by 4: $x = \frac{10}{4} = \frac{5}{2}$ .

Check: $4 \cdot \frac{5}{2} + 6 = 10 + 6 = 16$ . ✓

Example 14 video

Example 15: Solve $- 2 x + 11 = 5 x + 9$

Subtract $5 x$ from both sides: $- 7 x + 11 = 9$ .
Subtract 11 from both sides: $- 7 x = - 2$ .
Divide both sides by $- 7$ : $x = \frac{2}{7}$ .

Check — left side:

- 2 (\frac{2}{7}) + 11 = - \frac{4}{7} + \frac{77}{7} = \frac{73}{7}

Right side:

5 (\frac{2}{7}) + 9 = \frac{10}{7} + \frac{63}{7} = \frac{73}{7}

Both sides match. ✓

Example 15 video

Example 16: Solve $\frac{1}{6} (x + 8) + \frac{5}{6} (2 x - 1) = 3$

Example 16 video. When fractions sit outside parentheses, multiply both sides by the common denominator to clear them. (This trick works only for equations — not for expressions without an equal sign.) Both fractions have denominator 6, so multiply every term by 6.

6 \cdot \frac{1}{6} (x + 8) = (x + 8)

6 \cdot \frac{5}{6} (2 x - 1) = 5 (2 x - 1) = 10 x - 5

6 \cdot 3 = 18

So the equation becomes:

(x + 8) + (10 x - 5) = 18 \Rightarrow 11 x + 3 = 18 \Rightarrow 11 x = 15 \Rightarrow x = \frac{15}{11}

Check — left side:

\frac{1}{6} (\frac{15}{11} + 8) + \frac{5}{6} (2 \cdot \frac{15}{11} - 1) = \frac{198}{66} = 3

Right side is 3. ✓

Slope

The slope between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is:

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Example 17: Slope between $(- 1, 6)$ and $(7, 2)$

Example 17 video. Pick either point as the first; just stay consistent. Let $x_{1} = - 1, y_{1} = 6$ and $x_{2} = 7, y_{2} = 2$ .

m = \frac{2 - 6}{7 - (- 1)} = \frac{- 4}{8} = - \frac{1}{2}

Equation of a Line in $y = m x + b$ Form

You always need two things to write the equation of a line:

The slope. If provided, you’re done with step one. Otherwise, calculate it from two points.
One point on the line. Then use point-slope form:
$y - y_{1} = m (x - x_{1})$
(The $x$ and $y$ without subscripts remain as variables.) Then simplify and isolate $y$ to identify $m$ and $b$ .

Example 18: Equation of the line through $(2, - 5)$ and $(8, 1)$

Step 1 — calculate the slope with $x_{1} = 2, y_{1} = - 5, x_{2} = 8, y_{2} = 1$ :

m = \frac{1 - (- 5)}{8 - 2} = \frac{6}{6} = 1

Step 2 — plug into point-slope form:

y - (- 5) = 1 (x - 2) \Rightarrow y + 5 = x - 2 \Rightarrow y = x - 7

So $m = 1$ and $b = - 7$ .

Example 18 video

Math Placement Practice - Stage 1: Refresh and Learn

How to Use This Guide

Arithmetic

Multiplication and Division of Integers

Example 1: 436×52

Example 2: 624÷16

Fractions

Example 3: (56)2

Example 4: 78÷13

Example 5: 29+37

Example 6: Write 0.6 as a fraction in simplest terms

Absolute Value

Example 7: |−15+8|

Prealgebra and Algebra

Evaluate (Order of Operations)

Example 8: −16÷2(1+3)

Simplify the Expression

Example 9: 7x−3+8x+9

Example 10: −x2+7x−2+5x2+3x−6