# 1.1 Fractions: Defining Terms

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1.1 Fractions: Defining Terms
Fraction: part of a whole - example Numerator: number on top Denominator: number on bottom Proper Fraction: numerator is less than the denominator Improper Fraction: numerator is equal to or greater than the denominator

1.1 Problem Solving with Fractions
Mixed Number Consists of a whole number and a proper fraction – example

1.2 Changing the Form of a Fraction
Converting a mixed number to an improper fraction: Converting an improper fraction to a mixed number: Divide 9 into 35:

1.2 Changing the Form of a Fraction
Multiplying or dividing the numerator (top) and the denominator (bottom) of a fraction by the same number does not change the value of a fraction. Writing a fraction in lowest terms: Factor the top and bottom completely Divide the top and bottom by the greatest common factor

1.2 Changing the Form of a Fraction
A number can be divided evenly by: 2 – if the last digit is 0, 2, 4, 6, 8 3 – if the sum of the digits is divisible by 3 4 – if the last two digits form a number that is divisible by 4 5 – if the last digit is 0 or 5 6 – if the number is divisible by 2 and 3

1.2 Changing the Form of a Fraction
A number can be divided evenly by: 7 – double the last digit and subtract it from a number formed by the other digits. This number must be zero or divisible by 7 8 – if the last three digits form a number that is divisible by 8 9 – if the sum of the digits is divisible by 9 10 – if the last digit is 0

1.2 Changing the Form of a Fraction
A prime number can only be divided evenly by itself and the number 1 Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, etc. Factor trees can be used to factor a number to its prime factorization

1.2 Changing the Form of a Fraction – Factor Trees
18 6 3 2 3

Adding fractions with the same denominator: Subtracting fractions with the same denominator:

To add or subtract fractions with different denominators - get a common denominator. Using the least common denominator: Factor both denominators completely Multiply the largest number of repeats of each prime factor together to get the LCD Multiply the top and bottom of each fraction by the number that produces the LCD in the denominator

1.3 Adding and Subtracting Fractions – no common factors in denominator
Adding fractions with different denominators: Subtracting fractions with different denominators:

Try these:

1.4 Multiplying and Dividing Fractions
Multiplying fractions: Dividing fractions:

1.4 Multiplying and Dividing Fractions
Complex Fractions: The numerator, denominator or both are some sort of fraction (proper, improper, or mixed) Example:

1.4 Multiplying and Dividing Fractions
Try these:

2.1 Reading, Writing, and Rounding Decimals
Place value: the position of the number in relation to the decimal place What power of 10 does the 4 represent? What does the 8 represent? What about the 1?

2.1 Reading, Writing, and Rounding Decimals
Translating a decimal to words: In words: Twelve and thirty-two hundredths Translate the following:

2.1 Reading, Writing, and Rounding Decimals
Rounding a decimal: Look at the digit to the right of the place to which you are rounding If the digit is less than 5 all the digits to the right of the place you are rounding become zero If the digit is 5 or greater, the place you are rounding to is increased by 1 and all the digits to the right of the place you are rounding become zero Drop zeros to the right of the decimal place

2.1 Reading, Writing, and Rounding Decimals
Round to the nearest tenth: 5 is next to the tenths place so increase 4 by 1 to get Drop the zeros:

Write each number so that the decimal points are in a vertical line Add the numbers as if there were no decimal points. Place the decimal point in the answer in line with the other decimal points

Examples:

2.3 Multiplying and Dividing Decimals
To multiply decimals: Multiply the numbers as if there were no decimal points. Count the number of decimal places in each number and add them together Put that many decimal places in the answer Answer: 0.15

2.3 Multiplying and Dividing Decimals
To divide decimals: Write the numbers in long division format Move the decimal in the divisor to the right until you have a whole number Move the decimal in the dividend to the right the same number of places Divide as if the decimal points were not there Place the decimal in the answer just above the decimal in the dividend

2.3 Multiplying and Dividing Decimals
Example:

2.4 Converting Fractions and Decimals
Converting decimals to fractions: Converting fractions to decimals:

2.5 Converting Decimals and Percents
Write a decimal as a percent by moving the decimal point 2 places to the right and attaching a percent sign: Example:

2.5 Converting Decimals and Percents
Write a percent as a decimal by moving the decimal point 2 places to the left and removing the percent sign: Example:

2.6 Converting Fractions and Percents
Write a fraction as a percent by converting the fraction to a decimal and then converting the decimal to a percent: Example:

2.6 Converting Fractions and Percents
Write a percent as a fraction by first changing the percent to a decimal then changing the decimal to the fraction and reduce: Example:

Supplement: Chapter 1 1.1 Scientific Notation
Writing a number in scientific notation: Move the decimal point to the right of the first non-zero digit. Count the places you moved the decimal point. The number of places that you counted in step 2 is the exponent (without the sign) If your original number (without the sign) was smaller than 1, the exponent is negative. If it was bigger than 1, the exponent is positive

Supplement: Chapter 1 1.1 Scientific Notation
Converting to scientific notation (examples): Converting back – just undo the process:

Supplement: Chapter 1 1.1 Scientific Notation
Multiplication with scientific notation: Division with scientific notation:

Supplement: Chapter 1 1.2 Uncertainty in Measurements
Accuracy: correctness of a measurement Example: The statue of liberty is inches tall – the measurement is very precise but inaccurate Precision: degree of correctness Examples: 3.2 cm is more precise than 3 cm but less precise than 3.24 cm

Supplement: Chapter 1 1.2 Uncertainty in Measurements
Absolute error Measurement Absolute error 23 mg 0.5 mg 23.2 mg 0.05 mg 2.035 mg mg

Supplement: Chapter 1 1.2 Uncertainty in Measurements
Lower limit = measurement – absolute error Upper limit = measurement + absolute error Relative error:

Supplement: Chapter 1 1.3 Estimation
“” means “approximately equal to” Interval estimate: look at the first digit to get the interval Example: Low Estimate: = 1000 High Estimate: = 1300 The actual sum is between 1000 and 1300

Supplement: Chapter 1 1.3 Estimation
Rounding was covered in section 2.1 of the text and can be used to find an estimate Example – find an estimate by rounding to the tens place:  = An estimation is 1160