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Multiples of 3 and 9
An integer is divisible by 3 if the sum of its digits is divisible by 3. An integer is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits is 957 is 21, which is divisble by 3 but not by 9, so 957 is divisble by 3 but not by 9.
Multiples of 5 and 10
An integer is divisible by 5 if the last digit is 5 or 0. An integer is divisible by 10 if the last digit is 0. The last digit of 665 is 5, so 665 is a multiple of 5 but not a multiple of 10
The remainder is the whole number left over after divisions. 487 is 2 more than 485, which is a multiple of 5, so when 487 is divided by 5, the remainder will be 2.
To reduce a fraction to lowest terms; factor out and cancel all factors the numerator and denominator have in common
To add or subtract fractions, first find a common denominator, then add or subtract the numerator.
To multiply fractions, multiply the numerators and multiply the denominators
To divide fractions, invert the second one and multiply.
Mixed Numbers and Improper Fractions
To convert a mixed number to an improper fraction, multiply the whole number part by the denominator, then add the numerator. The result is the new numerator (over the same denominator).
To find the reciprocal of a fraction, switch the numerator and the denominator. The reciprocal of 7/3 is 3/7
One way to compare fractions is to re-express them with a common denominator. Another method is to convert them both to decimals.
Converting Fractions and Decimals
To convert a fraction to a decimal, divide the bottom into the top. To convert a decimal to a fraction, set the decimal over 1 abd multiply the numerator and denominator by 10 raised to the number of digits to the right of the decimal point.
To find a particular digit in a repeating decimal, note the number of digits in the cluster that repeats. If there are 2 digits in that cluster, then every second digit is the same. And so on.
Identifying the Parts and Wholes
The key to solving most fractions and percent story problems is to identify the part and the whole. Usually you'll find the part associated with the verb is/are and the whole associated with the word of.
- Whether you need to find the part, the whole, or the percent, use the same formula
- Part = Percent * Whole
Percent Increase and Decrease
To increase a number by a percent, add the percent to 100 percent, convert to a decimal, and multiply. To increase 40 by 25 percent, add 25 percent to 100 percent, convert 125 percent to 1.25 and multiply 40.
Finding the Original Whole
To find the original whole before a percent increase or decrease, set up an equation. Think of the result of a 15 percent increase
Combined Percent Increase and Decrease
To determine the combined effect of multiple percent increases and/or decreases, start with 100 and see what happens.
Setting Up a Ratio
To find a ratio, put the number associated with word of on top and the quantity associated with the word to on the bottom and reduce.
Part-to-Part Ratios and Part-to-Whole Ratios
If the parts add up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number in the original ratio over the sum of the numbers.
Solving a Proportion
To solve a proportion, cross multiply.
To solve a rates problem, use the units to keep things straight.
- Average Rate is not simply the average of the rates
- Average A per B = Total A/Total B
- Average Speed = Total Distance/ Total Time
To find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, don't just average the two speeds.
First figuew out the total distance and the total time
To find the average of a set of numbers, add them up and divide by the number of numbers.
- Average = Sum of the terms/ Number of terms
Average of Evenly Spaced Numbers
To find the average of evenly spaced numbers, just average the smallest and the largest.
Using the Average to Find the Sum
- Sum = (Average) * (Number of terms)
Finding the Missing Number
To find a missing number when you're given the average, use the sum.
Median and Mode
The median of a set of numbers is the value that falls in the middle of the set. The mode of a set of numbers us the value that appears most ofen.
Counting the Possibilities
The fundamental counting principle: If there are n ways a second event can happen, then there are m * n ways for the 2 events to happen
Probability = Favorable Outcomes/Total Possible Outcomes
Multiplying and Dividing
To multiply powers with the same base, add the exponents and keep the same base. To divide powers with the same base, subtract the exponents and keep the same base.
Raising Powers to Powers
To raise a power, multiply the exponents.
Simplifying Square Roots
To simplify a square root, factor out the perfect squares under the radical, unsquare them, and put the result in front.
You can add or subtract radical expressions when the part under the radicals in the same.
The product of square roots is equal to the square root of the quotient
Negative Exponent and Rational Exponent
To find the value of a number raised to a negative exponent, simply rewrite the number without the negative sign, as the bottom of a fraction with 1 as the numerator of a fraction
Determining Absolute Value
The absolute value of a number is the distance of the number from zero on the number line. Because absolute value is a distance, it is always positive.
Evaluating an Expression
To evaluate an algebraic expression, plug in the given values for the unknown and calculate according to PEMDAS.
To combine like terms, keep the variable part unchanged while adding or subtracting the coefficients.
To add or subtract polynomials, combine like terms.
To multiply monomials, multiply the coefficients and the variables separately.
To multiply binomials - use FOIL.
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