The University of Texas at El Paso
Tutoring and Learning Center
ACCUPLACER
MATH 0300
http://www.academics.utep.edu/tlc
32
MATH 0300
Page
Fractions 2
Fractions – Exercises 7
Fractions – Answers to Exercises 8
Decimal Numbers 10
Decimal Numbers – Exercises 12
Decimal Numbers – Answers to Exercises 13
Percents 14
Percents – Exercises 16
Percents – Answers to Exercises 17
Order of Operations 18
Order of Operations – Exercises 19
Order of Operations – Answers to Exercises 20
Real Numbers, the Number Line, Order Relations and Absolute Value 21
Real Numbers, the Number Line, Order Relations and Abs. Value – Exercises 23
Real Num., the Num. Line, Order Rel. and Abs. Value – Ans. to Exercises 24
Scientific Notation 25
Scientific Notation – Exercises 26
Scientific Notation – Answers to Exercises 27
Exponents 28
Radicals 31
Laws of Exponents and Radicals 33
Exponents and Radicals – Exercises 34
Exponents and Radicals – Answer to Exercises 35
32
FRACTIONS
A fraction is defined as a ratio of two numbers, where the number at the bottom cannot be equal to zero.
where
In a fraction the number at the top is called the numerator, and the number at the bottom is called the denominator.
There are two different kinds of fractions, proper and improper. In a proper fraction the numerator (the number at the top) is less than the denominator (the number at the bottom). In an improper fraction, the numerator is greater than the denominator.
Proper fraction: numerator < denominator Example:
Improper fraction: numerator > denominator Example:
When you have an improper fraction, it is not always right or recommended to leave it as your final answer. It is always best to change an improper fraction to a mixed number. A mixed number is a whole number and a fraction together.
Mixed number: C is a whole number and is a fraction
To change an improper fraction to a mixed number we need to divide the numerator by the denominator.
Let’s take our previous example of an improper fraction to make it into a mixed number:
Example:
When working with a fraction that contains large-value-numbers, it is always best to reduce the fraction to an equivalent fraction. An equivalent fraction is a fraction that has the same value but contains different numbers. To find an equivalent fraction, simply multiply the numerator and denominator by the same number.
Example: then
In the case of reducing a fraction, or simplifying the fraction as you may also call it, we need to divide the numerator and denominator by the same number.
Example: therefore
We have now found a fraction that is of the same value as the original one, but contains smaller digits and is easier to work with.
Now let’s get into the basic operations of fractions.
1. Addition or subtraction of fractions with the same denominator.
When you need to add or subtract fractions that contain the same denominator, all you need to do is to add or subtract the numerators, depending on the problem, and keep the same denominator.
Example: Addition:
Subtraction:
2. Addition or subtraction of fractions with different denominators.
To add or subtract fractions with different denominators, we must first find a common number between the two denominators making the problem easier to solve. This number is called least common denominator (LCD). Using the LCD makes the same denominator for all the fractions and the operation will be easier to perform.
Suppose we have the following problem:
To find the LCD, we must think of a common multiple for both denominators. Our denominators are 5 and 9 their multiples are as follow.
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 …
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 …
As you can see our LCD is 45 since it is the first multiple that both numbers have in common.
Now that we have found our LCD we can continue on with the operation. Rewrite both fractions to equivalent fractions with 45 as their denominator:
We now have an addition of fractions with the same denominator and are now able to perform the operation:
3. Multiplication of fractions.
In multiplication of fractions, multiply numerators with numerators and denominators with denominators.
4. Division of fractions.
To divide fractions we first need to invert the second fraction and then perform a multiplication of fractions.
5. Operations with whole numbers.
When dealing with whole numbers, we must first convert them into fractions. To do this simply put a 1 as the denominator and treat it as a fraction.
6. Changing fractions to decimals.
To change fraction into decimals we simply need to divide the numerator by the denominator.
Therefore:
7. Fraction proportions.
Proportions are equivalent fractions that are missing a number, either the numerator or the denominator. We use cross multiplication to find the missing number.
Example:
8. Word problems containing fractions.
When working with word problems that involve fractions, follow the procedures described previously to solve the problem.
Example: A rectangular piece of material 3 feet wide by feet long is cut into five equal strips. Find the length of each strip.
Find the length of each strip by cutting the piece length wise, so divide by 5:
Therefore, each piece will be feet.
FRACTIONS – EXERCISES
Reduce:
1. 2. 3.
4. 5. 6.
7. 8. 9.
10.
Find x:
11. 12. 13.
14. 15. 16.
Perform the indicated operations and simplify your answer:
17. 18. 19.
20. 21. 22.
23. 24. 25.
26. 27. 28.
FRACTIONS - ANSWERS TO EXERCISES
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
DECIMAL NUMBERS
1. Addition or subtraction of decimal numbers.
When adding or subtracting decimal numbers, the first step is to align the decimal point.
Example: Add 7.01 and 6.3 Add a 0 to align all numbers
Subtract 29.01 and 17.43
2. Multiplication of decimal numbers.
When multiplying decimals there is no need to align the point. However, the decimal point in the final answer will be placed according to the total number of places to the right of the decimal point.
Example: Multiply 1.25 and 3.2
2 digits to the right of the decimal point
1 digit to the right of the decimal point
Total of 3 digits to the right of the decimal point
Count three digits from right to left to place the decimal point
3. Division of decimal numbers.
When dividing decimals, the divisor must contain no decimal places. If it does, move the decimal place to the right until it becomes a whole number. The number of places you move the decimal point on the divisor will indicate the number of places you need to move your point in the dividend.
Example: Divide 117.525 by 2.5
4. Change decimals into fractions.
To change decimals into fractions the first step is to write the decimal number with no decimal point. This will be the numerator. The denominator will be 10, 100, 1000, etc, depending on the numerator. If it is a single digit number it will be 10, a double digit number will get 100 as the denominator, and so on.
Example: Change 0.41 into a fraction.
Step 1: Write the decimal number with no decimal point, do not write the
zero: 0.41 41 41 will be the numerator
Step 2: The denominator in this case will be 100, since 41 is a double
digit number:
Therefore:
5. Word problems containing decimals.
To solve word problems that contain decimals, follow the procedures described above depending on the type of problem.
Example: If you earn $526.35 and $64.52 is taken out on taxes, what is your total after-tax
earnings?
Subtract the amount of money taken out from your total amount to obtain the total after-tax earnings:
Amount after-tax.
DECIMAL NUMBERS - EXERCISES
Perform the indicated operations.
1. 76.19 + 19.2
2. 56.4 + 20.6
3. 197.64 – 129.37
4. 20.1 – 18.367
5.
6.
7.
8.
9.
10.
11. If one pound of apples costs $0.79, how much would 2 1/2 pounds cost?
12. If a 14 1/2 pound watermelon costs $2.99, how much does the watermelon cost per pound?
13. If you earn $35.25 on Saturday and $31.75 on Sunday, what is your total pay for the weekend?
14. If you earn $236.05 and $39.73 is taken out for taxes, what is your total after-tax earnings?
DECIMAL NUMBERS – ANSWERS TO EXERCISES
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11.
12. 13. 14.
PERCENTS
Percent means per cent, or per every hundred. In other words, percent means how much of the whole amount you have. A percentage can also be represented as a fraction whose denominator is 100.
Examples: 46% means or 46 out of 100.
10% means or 10 out of 100. It can also be reduced to.
1. Convert a percent to a fraction.
To convert a percentage into a fraction, divide the percentage number by 100, then reduce if necessary.
Examples: 32%
25%
2. Convert a percent to a decimal.
To convert a percentage into a decimal shift the decimal point in the percentage two places to the left.
Examples: 7%
6.38%
3. Convert a decimal to a percent.
To change a decimal into a percentage shift the decimal point two places to the right.
Examples: 0.36 36%
0.025 2.5%
4. Convert a fraction to a percent.
To convert a fraction into a percentage, first divide the numerator by the denominator and then change the decimal into a percentage.
Examples:
5. Problems involving percents.
When dealing with problems involving percentages, remember that “of” means multiplication and “is” means equals.
Examples: What is 15% of 260?
First convert the percent to decimal:
Then convert the words into mathematical terms:
or
Finally, solve the problem:
Answer: 15% of 260 is 39.
50 is what percent of 940?
In this case the percentage is the unknown. Remember, first convert the words into mathematical terms:
or
Solve the problem:
or 5.3%
PERCENTS – EXERCISES
1. 15% of 750 is what number?
2. 85% of what number is 255?
3. What is 18% of 350?
4. 75 is what percent of 300?
5. What percent of 450 is 250?
6. What number is 20% of 45?
7. 195 is 35% of what number?
8. 150 is what percent of 500?
9. What is 6.5% of 45?
10. 10.5% of what number is 21?
PERCENTS – ANSWERS TO EXERCISES
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
ORDER OF OPERATIONS
When dealing with a problem that contains different types of operations there are a series of steps that need to be followed.
1) Do the operations inside of parentheses. Then eliminate.
2) If there are any exponents present, simplify the numbers to eliminate of the exponents.
3) Next, do all multiplications and divisions present from left to right.
4) Finally, do all addition and subtractions from left to right.
Examples:
1. Solve:
First thing is to eliminate the parentheses,
Carry out the exponential operation the exponent,
Do the multiplication,
Do additions and subtractions from left to right
2. Solve:
3. Solve:
ORDER OF OPERATIONS – EXERCISES
Perform the indicated operations:
1.
2.
3.
4.
5.-
6.-
7.-
8.-
9.-
10.-
ORDER OF OPERATIONS – ANSWERS TO EXERCISES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
REAL NUMBERS, THE NUMBER LINE, ORDER RELATIONS, AND ABSOLUTE VALUE
When talking about numbers, there are different types of categories in which digits can be classified. The graphical representation of numbers is called the number line.
0 1 2 3 4 5
Integers are whole numbers; that is, numbers that contain no decimal point or numbers that are not in fraction form.
Example:
Integers are classified as positive or negative. Positive integers are numbers that contain no negative sign. These numbers are placed to the right of zero on the number line. Negative integers are numbers that contain a negative sign; these are placed to the left of zero on the number line. Zero is neither positive or negative.
Examples: Positive integers: 1, 2, 3, 4, 5, …
Negative integers:
Even integers are those integers which can be evenly divided by 2. Those integers which cannot be evenly divided by 2 are called odd integers.
Examples: Even integers: 2, 4, 8, 10, 12,
Odd integers: 3, 7, 15, 21, 25,
Prime numbers are integers that are greater than 1, but that are not divisible by any integer other than themselves and 1.
Examples: Prime integers: 2, 5, 7, 17, 19, 23, 29
When numbers are presented in the form of a fraction they are called rational numbers. When a number cannot be represented in fraction form this number is called an irrational number.
Examples: Rational numbers:
Irrational numbers:
When comparing any two real numbers, refer to the number line to see which number is greater or smaller. The number to the left is always the smaller number; while the number to the right is always the greater number of the two. This comparison is called order relations.
Examples: