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SAT Math
5. Advanced Arithmetic
Written by: Pranav I
Formatted by: Rithanya S
RATIOS AND PROPORTIONS
A Ratio Is a Comparison
- A ratio is a comparison between the quantities of ingredients you have in a mixture
- Ways to express ratios
- x/y
- the ratio of x to y
- x:y
Part, Part, Whole
- Ratios are a lot like fractions
- The parts of a ratio can be added to get the whole
- For example, the total number of ingredients in x:y is (x+y)
- NOTE: This is the simplest form
Ratio to Real
- Find the total number of parts
- Divide the total value by the total number of parts
- Multiply the parts in the ratio by the value per part
🔥 Warning
The test-writers will often combine ratios with charts and other diagrams- do not let them intimidate you!
- Read the question to avoid writing the right numbers in the wrong order
- 9:1 instead of 1:9, for example
Proportions Are Equal Ratios
- Some Digital SAT questions will contain two equal ratios from which one piece of information is missing.
Proportions: Advanced Principles
- Many proportion questions will have unit conversions- be sure to pay attention to the units and have the same units in both the numerators and the same units in both the denominators.
PERCENTAGES
Percentages Are Fractions
- Percent means “per 100” or “out of 100”
Converting Percentages to Fractions
- Put the percentage over 100 and reduce.
Converting Fractions to Percentages
- Divide the numerator by the denominator and multiply the quotient by hundred.
Converting Percentages to Decimals
- Move the decimal point two places to the left.
Converting Decimals to Percentages
- Move the decimal point two places to the right.
Percentage Increase or Decrease
- % change = (difference x 100) / (original)
What Percent of What Percent of What?
NOTE: Each successive increase or decrease is performed on the result of the previous one
🔥 Tip
Always handle percentage problems using Bite-Sized Pieces: one piece at a time
What Percent of What Percent of… Yikes!
- When you are not given actual numbers to work with in successive percentage questions, you need to plug in some numbers.
AVERAGES
What Is an Average?
- The average, also called the arithmetic mean, of a set of n numbers is simply the sum of all the numbers divided by n
- Total = Average x Number of things or T = AN
Averages: Advanced Principles
- In more difficult average questions, use the T=AN formula more than once
- To find the numbers to average when charts are given, look them up on the graphic provided
- WATCH OUT FOR MISMATCHED UNITS!
- The mean of a list is not likely to change dramatically with the removal of the numbers at the extremes
WHAT IS A MEDIAN?
- The median of a list of numbers is the number that is exactly in the middle of the list when the list is arranged from smallest to largest, on a number line (ascending order)
- In a set containing an even number of items, take the average of the two middle numbers to get the median
WHAT IS A MODE?
- The mode of a group of numbers is the number in the list that appears most often
- MODE is the Most Occuring Data Entry
- Questions will most often combine mode with other concepts like mean, median or range
WHAT IS A RANGE?
- The range of a list of numbers is the difference between the greatest number in the list and the least number in the list
WHAT IS A MARGIN OF ERROR?
- A margin of error is a range of percentages
- Gives a range for random sampling errors in a survey or poll
- Indicates how much the result might change if the poll were repeated or if the entire population were asked instead of a random sample
WHAT IS A FREQUENCY TABLE?
- A frequency table is a table to show how frequently something happens
- Gives you ways of finding the mean, median and mode without writing down the whole list
WHAT IS STANDARD DEVIATION?
- The standard deviation shows the spread of a group of numbers
- When the numbers are more spread out around the mean, the standard deviation is greater
- When the numbers are clumped closer together around the mean, the standard deviation is smaller
PROBABILITY
- Probability is the mathematical expression of the likelihood of an event
- Probability of x
- (Number of outcomes that give you what you want) / (Total number of possible outcomes)
- The largest a probability can be is 1; a probability of 1 indicates total certainty
- The smallest a probability can be is 0, meaning that it’s something that cannot happen
- P(something will not happen) = 1 – P(something will happen)
RATES
- Rates is a concept related to averages
- D = RT (distance) and W = RT (work)
- Here, R is the Rate and T is the Time
