Math Knowledge
Ratios, Percents & Data
Ratios, percents, probability, and basic statistics are all ways of comparing quantities and interpreting information. For ASVAB Math Knowledge, you should be comfortable comparing quantities with ratios, solving simple proportions, finding unit rates, converting between fractions, decimals, and percents, calculating percent change, finding basic probabilities, calculating mean, median, mode, and range, and reading simple frequency data. The most important skill is identifying what the numbers represent before calculating.
Section 1
Ratios
A ratio compares two quantities. The comparison 3 to 5 can be written 3:5 or 35.
Order Matters
If a class has 8 boys and 12 girls, the ratio of boys to girls is 8:12, while the ratio of girls to boys is 12:8. Those are different comparisons.
Simplifying Ratios
Simplify ratios like fractions. For 12:18, divide both quantities by 6 to get 2:3.
Part-to-Part Ratios
A part-to-part ratio compares one group with another. If a box contains 4 red balls and 6 blue balls, red to blue is 4:6=2:3.
Part-to-Whole Ratios
A part-to-whole ratio compares one group with the total. With 4 red and 6 blue balls, there are 10 total, so red to all balls is 4:10=2:5.
Rules and formulas
- Keep quantities in the requested order.
- Divide both parts by the same common factor.
- For part-to-whole ratios, calculate the total first.
Example 1
A team has 12 experienced workers and 8 new workers. Find the ratio of experienced workers to new workers in simplest form.
Solution
- 1Write the ratio: 12:8.
- 2The GCF of 12 and 8 is 4.
- 3Divide both parts by 4: 12÷4=3 and 8÷4=2.
Why: A ratio is simplified by dividing both quantities by the same common factor, just like simplifying a fraction.
Example 2
A container has 6 green parts and 9 yellow parts. Find the ratio of yellow parts to green parts.
Solution
- 1The requested order is yellow:green.
- 2Write 9:6.
- 3Divide both numbers by 3: 9÷3=3 and 6÷3=2.
Why: The order of a ratio must match the order in the question. Yellow comes first, so 9 must come first.
Example 3
A bag contains 8 red marbles and 12 blue marbles. Find the ratio of red marbles to all marbles.
Solution
- 1Find the total: 8+12=20.
- 2Compare red marbles with the total: 8:20.
- 3Divide both numbers by 4: 8÷4=2 and 20÷4=5.
Why: This is a part-to-whole ratio, so the second number represents all marbles, not only the blue marbles.
Section 2
Proportions & Unit Rates
A proportion states that two ratios are equal. For example, 23=69. A unit rate compares a quantity with exactly one unit.
Solving Proportions
Use cross multiplication. If ab=cd, then ad=bc.
For 34=x20, cross multiply to get 3×20=4x, so 60=4x and x=15.
Unit Rates
Unit rates describe miles per one hour, dollars per one item, or words per one minute.
A car traveling 180 miles in 3 hours has unit rate 180÷3=60 miles per hour.
Finding Better Value
Use unit prices to compare offers. Eight items for $16 cost $2 each, while six items for $15 cost $2.50 each. The first package has the lower unit cost.
Rules and formulas
- For ab=cd, cross multiplication gives ad=bc.
- A unit rate has a denominator of one unit.
- For price comparisons, calculate cost per item.
Example 1
Solve 45=x25.
Solution
- 1Cross multiply: 4×25=5x.
- 2100=5x.
- 3Divide both sides by 5: x=20.
Why: A proportion contains equivalent ratios. Cross multiplication creates an equation that can be solved for the missing value.
Example 2
A machine produces 144 parts in 6 hours. How many parts does it produce per hour?
Solution
- 1Find the unit rate.
- 2144÷6=24.
Why: A unit rate tells how much occurs for exactly one unit, so total parts are divided by the number of hours.
Example 3
Store A sells 12 bottles for $18. Store B sells 10 bottles for $16. Which store has the lower cost per bottle?
Solution
- 1Store A: $18÷12=$1.50 per bottle.
- 2Store B: $16÷10=$1.60 per bottle.
- 3Compare: $1.50<$1.60.
Why: Comparing unit prices puts both offers on the same basis: the cost of one bottle.
Section 3
Percents
Percent means per hundred. For example, 35%=35100=0.35.
Percent, Decimal, and Fraction Conversions
To change a percent to a decimal, move the decimal point two places left: 42%=0.42.
To change a decimal to a percent, move it two places right: 0.68=68%.
For a fraction such as 34, divide to get 0.75, then convert to 75%.
Finding a Percent of a Number
Convert the percent to a decimal and multiply. For 25% of 80, calculate 0.25×80=20.
Percent Increase and Decrease
Use changeoriginal×100%. The denominator is the original amount.
A rise from $50 to $60 is an increase of $10, so 1050=0.20=20%.
Finding the Original Value
If a value is $80 after a 20% decrease, then 80% remains. Write 0.80× original=80, so the original is $100.
Rules and formulas
- Percent to decimal: move two places left.
- Decimal to percent: move two places right.
- Percent of a number: decimal form times the number.
- Percent change: changeoriginal×100%.
Example 1
Find 35% of 240.
Solution
- 1Convert 35% to a decimal: 35%=0.35.
- 2Multiply: 0.35×240=84.
Why: Percent means a portion of a whole, so converting the percent to a decimal and multiplying finds that portion of 240.
Example 2
A value increases from 80 to 100. Find the percent increase.
Solution
- 1Find the increase: 100-80=20.
- 2Divide by the original value: 2080=0.25.
- 3Convert to a percent: 0.25=25%.
Why: Percent change compares the amount of change with the original amount, not the new amount.
Example 3
A jacket costs $72 after a 20% discount. Find the original price.
Solution
- 1A 20% discount means 80% remains.
- 2Convert: 80%=0.80.
- 3Let the original price be x: 0.80x=72.
- 4Divide by 0.80: x=90.
Why: If 20% was removed, the sale price represents 80% of the original price.
Section 4
Probability
Probability measures how likely an event is to occur. Use P(event)=favorable outcomestotal possible outcomes. Probability may be written as a fraction, decimal, or percent.
Probability Range
Probability ranges from 0 to 1, or from 0% to 100%. A probability of 0 is impossible; a probability of 1 is certain.
Simple Probability
A bag with 3 red and 2 blue balls has 5 total balls, so P(red)=35.
Complement Probability
The complement is the probability an event does not happen. Use P(not A)=1-P(A).
If P(rain)=0.30, then P(no rain)=1-0.30=0.70.
Independent Events
For independent events, one event does not affect the other. To find the probability that both occur, multiply.
For two heads in two fair coin flips, calculate 12×12=14.
Rules and formulas
- Probability is favorable outcomes divided by total outcomes.
- Complement: 1-probability.
- For independent events that both occur, multiply probabilities.
Example 1
A bag contains 5 red balls, 3 blue balls, and 2 green balls. Find the probability of choosing a blue ball.
Solution
- 1Find the total: 5+3+2=10.
- 2There are 3 favorable blue outcomes.
- 3Use favorable over total: 310.
Why: There are three favorable outcomes out of ten total possible outcomes.
Example 2
The probability that a machine fails a test is 18. Find the probability that it does not fail.
Solution
- 1Use the complement: 1-18.
- 2Rewrite 1 as 88.
- 3Subtract: 88-18=78.
Why: Either the machine fails or it does not fail, so the two probabilities must add to 1.
Example 3
A fair coin is flipped and a fair six-sided die is rolled. Find the probability of heads and a 6.
Solution
- 1P(heads)=12.
- 2P(rolling a 6)=16.
- 3The events are independent, so multiply: 12×16=112.
Why: For independent events that must both occur, multiply their individual probabilities.
Section 5
Mean, Median, Mode & Range
These measurements summarize sets of data. Mean is the arithmetic average, median is the middle ordered value, mode is the most frequent value, and range measures the spread from smallest to largest.
Mean
Use Mean=sum of valuesnumber of values. For 4, 6, and 8, the sum is 18 and the mean is 183=6.
Median
Arrange values from smallest to largest and find the middle. If there are two middle values, average them.
For 2, 4, 8, 10, the median is 4+82=6.
Mode
The mode is the value appearing most often. A data set can have one mode, more than one mode, or no mode.
Range
Use Range=largest-smallest. For 4, 7, 11, 15, the range is 15-4=11.
Rules and formulas
- Mean: add and divide.
- Median: order values and find the middle.
- Mode: most frequent value.
- Range: largest minus smallest.
Example 1
Find the mean of 72, 80, 88, and 100.
Solution
- 1Add the values: 72+80+88+100=340.
- 2There are 4 values.
- 3Divide: 340÷4=85.
Why: The mean distributes the total value equally among the number of observations.
Example 2
Find the median of 14, 6, 20, 10, and 8.
Solution
- 1Put the numbers in order: 6, 8, 10, 14, 20.
- 2There are five values.
- 3The middle value is 10.
Why: The median is based on position, so values must be ordered before finding the middle.
Example 3
For 3, 7, 7, 9, 12, and 14, find the mode and range.
Solution
- 1The value appearing most often is 7, so Mode=7.
- 2The largest value is 14 and the smallest is 3.
- 3Range=14-3=11.
Why: The mode identifies the most frequent value, while the range measures the distance between the largest and smallest values.
Section 6
Frequency Tables & Simple Data
A frequency table shows how often different values occur. Instead of listing every observation, it gives each value and the number of times that value appears.
Reading a Frequency Table
If score 1 has frequency 2, score 2 has frequency 3, and score 3 has frequency 1, the full data set is 1, 1, 2, 2, 2, 3.
Total Observations and Mode
Add the frequencies to find the number of observations. The value with the greatest frequency is the mode.
Mean From a Frequency Table
Multiply each value by its frequency, add the products, and divide by total frequency.
For values 1, 2, 3 with frequencies 2, 3, 1: total value=1(2)+2(3)+3(1)=11, total observations=6, and mean=116.
Missing Value From a Known Mean
Use Total=Mean× Number of Values, then subtract the known values.
Fundamental Counting Principle
If one choice can be made in a ways and another independent choice in b ways, there are a× b combinations.
Three shirts and two pairs of pants make 3×2=6 outfits.
Example frequency table
| Score | Frequency |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 1 |
Rules and formulas
- Total observations: add frequencies.
- Mode: value with the greatest frequency.
- Mean: sum(value× frequency)total frequency.
- Counting principle: multiply independent numbers of choices.
Example 1
Scores 1, 2, 3, and 4 appear 2, 4, 3, and 1 times. How many total observations are there?
Solution
- 1Add the frequencies.
- 22+4+3+1=10.
Why: Frequency tells how many times each value appears, so adding all frequencies gives the total number of observations.
Example 2
Value 2 appears 2 times, value 4 appears 3 times, and value 6 appears 1 time. Find the mean.
Solution
- 1Multiply each value by its frequency: 2×2=4, 4×3=12, and 6×1=6.
- 2Add the products: 4+12+6=22.
- 3Add the frequencies: 2+3+1=6.
- 4Divide: Mean=226=113≈3.67.
Why: A frequency table represents repeated values, so each value must first be multiplied by the number of times it occurs.
Example 3
A restaurant offers 4 sandwich choices and 3 drink choices. How many one-sandwich, one-drink combinations are possible?
Solution
- 1There are 4 sandwich choices and 3 drink choices.
- 2Use the Fundamental Counting Principle.
- 34×3=12.
Why: Each sandwich can be paired with each drink, so the numbers of independent choices are multiplied.
Quick Review
Ratios
- A ratio compares quantities.
- Order matters.
- Part-to-part compares one group with another.
- Part-to-whole compares one group with the total.
- Simplify ratios like fractions.
Proportions & Unit Rates
- A proportion contains equal ratios.
- Cross multiplication can solve a missing value.
- A unit rate is an amount per one unit.
- For price comparisons, compare cost per item.
Percents
- Percent means per hundred.
- Percent to decimal: move two places left.
- Decimal to percent: move two places right.
- Percent of a number: multiply by decimal form.
- Percent change: changeoriginal×100%.
Probability
- Probability: favorable outcomes divided by total outcomes.
- Complement: 1-probability.
- For independent events that both occur, multiply probabilities.
Mean, Median, Mode & Range
- Mean: add and divide.
- Median: middle value after ordering.
- Mode: most frequent value.
- Range: largest minus smallest.
Frequency & Data
- Total observations: add frequencies.
- Mode: value with the largest frequency.
- Mean: sum(value× frequency)total frequency.
- Counting Principle: multiply independent numbers of choices.