Prealgebra/Unit Conversion: Workbook Examples/Practice

jtrude

1 Prealgebra/Unit Conversion: Workbook Examples/Practice

1.3 Order of Operations

When simplifying expressions, it is important to do so in the correct order. Consider the problem 2 + 5 ⋅ 3 done two different ways:

Method 1: Add first	Method 2: Multiply first
Add: 2 + 5 ⋅ 3	Multiply: 2 + 5 ⋅ 3
Multiply: 7 ⋅ 3	Add: 2 + 15
Solution: 21	Solution: 17

The previous example illustrates that if the same problem is done two different ways, it will result in two different solutions. However, only one method can be correct. It turns out the second method is the correct one.

Key Takeaways: Order of Operations

1st Brackets (Grouping Symbols); Work inner to outer; Distribute as needed

2nd Exponents and Roots

3rd Multiplication and Division (Left to Right)

4th Addition and Subtraction (Left to Right)

Key Takeaways: Exponents

The above example also illustrates an important point about exponents:

Exponents are only considered to be on the number they are attached to.
This means that, in the expression −4², only the 4 is squared, giving us −(4²) or −16.
But when the negative is in parentheses, such as in (−5)², the negative is part of the number and is also squared, giving a positive solution of 25.

1.3 Examples/Practice Problems

Reduce and solve the following expressions. Show all steps.

1.) $(-6 \div 6)^3$

2.) $5(-5 + 6) \cdot 6^2$

3.) $7 - 5 + 6$

4.) $(-2 \cdot 2^3 \cdot 2) \div (-4)$

5.) $(-7 - 5) \div [-2 - 2 - (-6)]$

6.) $\dfrac{5^2+(-5)^2}{| 4^2-2^5| -2 \cdot 3}$

7.) $\dfrac{-9 \cdot 2 - (3 - 6)}{1-(-2+1)-(-3)}$

1.4 Properties of Algebra

When doing algebra, it is common not to know the value of the variables. In this case, simplify where possible and leave any unknown variables in the final solution. One way to simplify expressions is to combine like terms.

Like terms are terms whose variables match exactly, exponents included. Examples of like terms would be $3xy$ and $-7xy,$ $3a^2b$ and $8a^2b,$ or −3 and 5. To combine like terms, add (or subtract) the numbers in front of the variables and keep the variables the same.

1.4 Examples/Practice Problems

Reduce and combine like terms.

1.) $4b + 6 + 1 + 7b$

2.) $1 - 10n - 10$

3.) $-2(n + 1)$

4.) $4v - 7(1 - 8v)$

5.) $-4k^2 - 8k(8k + 1)$

6.) $-2r(1 + 4r) + 8r(-r + 4)$

7.) $(7a^2 + 7a) - (6a^2 + 4a)$

1.6 Unit Conversion Word Problems

One application of rational expressions deals with converting units. Units of measure can be converted by multiplying several fractions together in a process known as dimensional analysis.

1.6 Examples/Practice Problems

Use dimensional analysis to perform the indicated conversions.

1.) 7 miles to yards

2.) 234 oz to tons

3.) 11.2 mg to grams

4.) 1.35 km to centimetres

5.) 4.5 ft² to square yards

6.) 3.5 mph (miles per hour) to feet per second

7.) On a recent trip, Jan travelled 260 miles using 8 gallons of gas. What was the car’s miles per gallon for this trip? Kilometres per litre?

8.) You are buying carpet to cover a room that measures 38 ft by 40 ft. The carpet cost $18 per square yard. How much will the carpet cost?

1.7 Activities

Problem 1

There are four known solutions to the following math puzzle, in which you can move just one line to fix the equation. How many solutions can you find?

Problem 2

Are the following statements true?

Letters A, B, C and D do not appear anywhere in the spellings of 1 to 99

Letter D appears for the first time in “hundred”

Letters A, B and C do not appear anywhere in the spellings of 1 to 999

Letter A appears for the first time in “thousand”

Letters B and C do not appear anywhere in the spellings of 1 to 999,999,999

Letter B appears for the first time in “billion”

Letter C does not appear anywhere in any word used to count in English