# Survival Skills for Quantitative Courses Algebra & Arithmetic

SURVIVAL SKILLS

ALGEBRA & ARITHMETIC

Scientific Notation

Logarithms

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# Using Positive and Negative Numbers

• Adding a negative number = Subtracting a positive number

4 + (-3) = 4 - 3 = 1

• Subtracting a negative number = Adding a positive number

4 - (-3) = 4 + 3 = 7

• Multiplying a negative number and a positive number gives a negative result

4 x (-3) = -12

• Multiplying a negative number and a negative number gives a positive result

(-4) x (-3) = 12

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# Manipulating Mathematical Expressions

## or How to solve for x

RULE
Any operation my be performed on an equation, provided the same operation is applied to both sides of the equals sign.

EXAMPLE
Given the equation:
2x + 4y = 12

Through mathematical manipulations, this expression can be transformed into one which defines x in terms of y.

2x + 4y = 12

2. Examine the formula. What changes need to be made to obtain an expression for x in terms of y? In this case:

• Subtract 4y from both sides:

2x + 4y - 4y = 12 - 4y
2x = 12 - 4y

• Divide both sides by 2:

2x/2 = (12 - 4y)/2

• Determine the final formula:

x = 6 - 2y

If the value of y is known, the value of x may be determined using this formula. For example, if y is 2, the value of x is 2.

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Chemistry problems are often solved using formulas. Generally, the formula is manipulated to give the appropriate expression for the problem in question and then the given values are substituted into the formula. The area of a rectangle is found by multiplying the lengths of the base and height together. The formula for the area of a rectangle is:

A=bh

If the base of a rectangle is 10 cm and the height is 4 cm the area is given by:

A = 10 cm x 4 cm

A = 40 cm

Note: the units multiply along with the numbers. cm x cm = cm2

## Manipulating formulas--I

If the area and the base of a rectangle are given, the height can be found by manipulating the original formula:

A=bh

2. Examine the formula. What changes need to be made to obtain an expression for the length of the height (h) in terms of the other variables (Area (A) and length of base (b))?

Divide both sides by the length of the base b:

A/b = (bh)/b
3. Determine the final formula:

A/b = h

If the area of a rectangle is 36 cm2 and the base is 6 cm, the height can be found using this new formula:

h = A/b = 36cm2/6cm = 6 cm

## Manipulating formulas--II

Some formulas require more manipulation but the basic procedure is the same. Consider the formula for the area of a circle:

If the area of a circle is known, the radius can be found using the forumla for area:

A = r 2

2. Examine the formula. What changes need to be made to obtain an expression for the radius (r) in terms of the other variable (Area, (A))? In this case, a constant is also involved.

• Divide both sides by :

A/ = (r 2)/

A/ =r 2

• Take the square root of both sides:

√ (A/) = √ (r 2)

3. Determine the final formula:

r = √ (A/)

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