Download Review of Basic Math Concepts

Basic Math Concepts Needed for Science
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BEDMAS
Scientific Notation
Solving for an unknown variable
Ratio comparisons
Significant Digits and Rounding
Dimensional Analysis or The Factor-Label Method
NOTE: When doing ANY calculation in science, it is ALWAYS required to include the units. The
units should be included when listing your givens, in your intermediate calculations, in your answer
and in your therefore statement!!! If units are not included you will lose marks.
NOTE: If a question which requires mathematical calculations is asked using words, then the answer
must be given using words as a therefore statement. If you are unsure if you need a therefore
statement you should ask. Questions which require therefore statements will also have marks given
for having a therefore statement.
BEDMAS
 When solving a mathematical equation with more than one operation you must solve the
question in the following order; Brackets, Exponents, Division or Multiplication, Addition or
Subtraction
Scientific Notation
 Scientists have developed a shorter method to express very large numbers. This method is
called scientific notation. Scientific Notation is based on powers of the base number 10.
 The number 123,000,000,000 in scientific notation is written as 1.23 x 1014
 The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less
than 10.
 The second number is called the base . It must always be 10 in scientific notation. In the
number 1.23 x 1011 the number 11 is referred to as the exponent or power of ten.
 When multiplying or dividing a number in scientific notation by a number which is not, the
answer must be in scientific notation
 Sample Question: 4.2 mol x (6.02x1023) atoms/mol = 2.5x1024 atoms
Solving for an unknown variable
 When using a formula to solve a problem you will often be given several values, and asked
to solve for the unknown variable.
 Example : If n = m/M, when n = 2.3mol and m = 4.567g what if the value of M?
 Solution: If n = m/M, then M = m/n. Substituting for the values we know gets M=4.567g /
2.3mol or M = 2.0 g/mol
Ratio comparisons
 The easiest way to compare ratios is to make them into a question which solves for an
unknown variable
 Example: If H2O and CO2 occur in a ratio of 2:5. If you have 3.7 moles of H2O, how many
moles of CO2 do you have?
 Solution: 2/5 = 3.7/x is the equation which can be developed. Solving for x, or moles of CO 2,
gives you the correct answer of 9.2 mol of CO2.
Significant Digits (Not only for Physicists!)
The Six Rules of Significant Digits
1. All nonzero digits are significant.
For example: 457 cm (three significant digits) and 0.25 g (two significant digits)
2. Zeros between nonzero digits are significant.
For example: 1005 kg (four significant digits) and 1.03 cm (three significant digits)
3. Zeros to the left of the first nonzero digit in a number are not significant; they merely indicate
the position of the decimal point.
For example: 0.02 g (one significant digit) and 0.0026 cm (two significant digits)
4. When a number ends in zeros that are to the right of the decimal point, they are significant.
For example: 0.0200 g (three significant digits) and 3.0 cm (two significant digits)
5. When a number ends in zero that are not to the right of a decimal digit, the zeros are not
necessarily significant.
For example: 130 cm (two or three significant digits); 10,300 g (three, four, or five significant
digits). The way to remove this ambiguity is to use scientific notation.
10,300 g
1.03 x 104
1.030 x 104
1.0300 x 104
(three significant digits)
(four significant digits)
(five significant digits)
6. Numbers obtained from counting are exact, so all digits are significant (counted as infinite
number in calculations involving multiplication and division)
Rounding
 If the digit after the one you want is greater than 5, then round up
For example: To obtain 2 significant digits: 3.47 rounds to 3.5 and 3.494 rounds to 3.5
 If the digit after the one you want is less than five then the preceding number stays the same
For example: To obtain 2 significant digits: 3.44 rounds to 3.4 and 3.449 rounds to 3.4
 If the single digit after the one you want is 5, round to the closest even number
For example: To obtain 2 significant digits: 2.55 is rounded to 2.6 and 2.25 is rounded to 2.2
Significant Digits in Calculations
Addition and Subtraction
The sum should have the same number of significant digits after the decimal as the least precise
value in the numbers being added.
e.g
2.345
2.336
2.2
6.881
 three decimal places
 three decimal places
 one decimal place
 round off to 6.9
4 kg
+ 43 g
is rewritten
4000 g  zero decimal place
+ 43 g  zero decimal place
4043 g  stays 4043 g
This is the precision rule.
Your answer cannot be any more precise than the least precise measuring instrument used.
Multiplication and Division
The answer has the same number of significant digits as the least number of significant digits
used in the calculations.
e.g. 21.3 cm = 16.384 615 cm  round off to16 cm
1.3 cm
e.g. (6.221 cm)(5.2 cm) = 32.3492 cm2  round off to 32 cm2
This is called the certainty rule.
Rounding during calculations
If a calculation has multiple steps, do not round until you have arrived at the final answer.
Problem Solving in Chemistry – Dimensional Analysis or The Factor Label Method
 The factor-label method is a logical and consistent way of converting a quantity in one unit into
the equivalent quantity in another unit
Example: Determine how many seconds are in 3.52 minutes
Step 1: Identify Key Value and Conversion Factor Equation
3.52 minutes (key value)
1 minute = 60 seconds (conversion factor equation or equality)
Step 2: Identify Required Value
The required value is number of seconds in 3.52 minutes.
Step 3: List Possible Conversion Factors
Using the conversion factor equation (equality), the possible conversion factors are
1 min
60s
 1 OR
1
60s
1 min
These fractions are equal to 1 because, in both cases, the numerators and denominators
are of equal value: 60 s is the same length of time as 1 min, and vice versa. All conversion factors
equal 1.
The only difference is that one fraction is inverted when compared to the other fraction.
The conversion factor you use in the solution to a problem depends on the units of the given value.
Choose the form of the conversion factor whose denominator has the same units as the
given value. Since multiplying by a conversion factor is like multiplying by 1, only the units change.
Step 4: Substitute Values into Solution Equation, and Solve
Required Value =
Seconds
=
Key Value
3.52 min
Conversion Factor
60 s
1 min
= 211.1 s (calculator)
= 211 s (applying significant digit rules – 3.52 min has 3 significant digits)
NOTE: Use as many conversion factors as required.
Scientific Notation Practice
1. Express each of the following numbers in scientific notation (exponential notation).
a. 10 000
f. 500 000 000 000 000
b. 0.000 1
g. 0.000 000 08
c. 1 000 000 000
h. 0.000 000 000 000 9
d. 0.000 000 000 000 001
i. 7 000
e. 400 000
j. 0.004
2. Express each of the following numbers in exponential notation.
a. 67 000
f. 0.001 000 045
b. 890 000 000
g. 0.003 880 908 3
c. 0.000 004 5
h. 45 887 950
d. 0.000 000 000 000 009 8
i. 36 000 000 000 000
e. 805 000
j. 0.000 000 000 000 015 05
3. Write out each of the following numbers in the long form.
a. 8.2 X 103
f. 3.86 X 106
b. 1.5 X 10-6
g. 6.892 X 10-3
c. 1.775 X 107
h. 9.035 X 10-10
d. 7.065 X 10-5
i. 5.425 X 102
e. 8.900 X 10-8
j. 2.662 X 10-1
4. Use scientific notation (exponential notation) to find the product of each of the following
a.
30 000 X 7 000
f.
10 000 000 X 0.000 005
b.
0.000 5 X 0.003
g.
3 000 X 0.000 4
c.
400 000 X 50 000
h.
0.000 000 000 05 X 0.003
d.
0.000 000 06 X 0.000 4
i.
0.000 000 03 X 7 000 000
e.
0.000 007 X 80 000
j.
900 000 X 0.000 000 009
Significant Digits and Rounding Practice
A. Indicate the number of significant figures then round each to the number of significant figures indicated.
For example:
1.234
has ______4___ significant figures and, rounded to
2
significant figures, is __1.2_
1.
0.6034
has __________ significant figures and, rounded to
2
significant figures, is ______
2.
12,700
has __________ significant figures and, rounded to
2
significant figures, is ______
3.
12,700.00 has __________ significant figures and, rounded to
1
significant figures, is ______
4.
0.000983 has __________ significant figures and, rounded to
2
significant figures, is ______
5.
123342.9 has __________ significant figures and, rounded to
5
significant figures, is ______
6.
6.023 x 1023has __________ significant figures and, rounded to
2
significant figures, is ______
7.
.005600
1
significant figures, is ______
8.
10000.5006 has __________ significant figures and, rounded to
5
significant figures, is ______
9.
2.0 x 10-3 has __________ significant figures and, rounded to
1
significant figures, is ______
10.
3.456110 has __________ significant figures and, rounded to
3
significant figures, is ______
has __________ significant figures and, rounded to
B. Given calculations with the calculator answer, write the answers with the appropriate number of significant figures.
Example:
6.00 x 3.00
= 18
The answer should be ______18.0_____
1.
23 + 46
= 69
The answer should be _______________
2.
23.0 + 46.0
= 69
The answer should be _______________
3.
253 + 345.8
= 598.8
The answer should be _______________
4.
56 – 35
= 21
The answer should be _______________
5.
56.00 – 35.0
= 21
The answer should be _______________
6.
46 x 12
= 552
The answer should be _______________
7.
3.24 x 5.63
= 18.2412
The answer should be _______________
8
(2.355 + 2.645) x 10.00
= 50
The answer should be _______________
9
654  28
= 23.35714286
The answer should be _______________
10.
0.024 x 0.063
= 1.512 x 10-3
The answer should be _______________
Solving for an unknown variable.
Remember that your answer must have the correct number of significant digits.
1) 8x = 67
2) 92 = 4x
3) 84.38 = 23x
4) 95847x = 493
5)
6.87 9.5

x
56.1
6) 4.236 
5.23
x
Using THE FACTOR-LABEL METHOD, convert the following…
1. Convert 57 mL to its equivalent in L.
2. Convert 15.9 mm to its equivalent in km.
3. Convert 35 kg to its equivalent in g.
4. Convert a speed of 88 m/s to its equivalent in cm/s.
5. Convert a speed of 73.5 km/h to its equivalent in m/s.
6. The density of mercury metal is 13.6 g/mL. What is the mass of 3.55 mL of the metal?