GED Math: Data, Statistics, and Probability
Data can be shown in different ways, such as a decimal, a fraction, a percent, a table, a graph, or a diagram, and the same proportion keeps its size no matter how it is written. To compare two values fairly, convert them to the same format, and a numerical format (decimals or percents) is usually the easiest. Typical steps: read the value from a graph or table, write it as a ratio or rate (the slope of a line, rise ÷ run, gives a rate such as miles per hour per second), turn it into a decimal or percent, and then compare. When asked how close a ratio is to a target, compare the differences from the target.
1. Reading Tables and Pictographs
Data are facts and numbers collected so we can answer questions and make decisions. A table organizes data in rows and columns, and a pictograph uses small pictures or symbols; a key tells you what one symbol is worth. Before you calculate, read every label (title, column heads, units, and the key), then use only the information the question needs. Tables often hold extra columns that you can ignore. Many questions combine data reading with earlier skills such as percent of change, ratios, and rates (for example, people per square mile is population divided by area).
Worked Example: Use the county table. (a) Which county had the greatest percent increase in population from 2000 to 2010? (b) In a pictograph where each symbol is 40 books, a branch has 5 1/2 symbols. How many books is that?
| County | 2000 pop. | 2010 pop. | Area (sq mi) |
|---|---|---|---|
| Adams | 8,400 | 10,500 | 700 |
| Bell | 21,000 | 19,950 | 350 |
| Cook | 5,500 | 5,830 | 1,100 |
| Davis | 60,000 | 66,000 | 440 |
(a) Only Adams, Cook, and Davis grew. Adams: (10,500 − 8,400) ÷ 8,400 = 2,100 ÷ 8,400 = 25%. Cook: 330 ÷ 5,500 = 6%. Davis: 6,000 ÷ 60,000 = 10%. Adams has the greatest percent increase. The area column is not needed.
(b) 5.5 × 40 = 220 books.
1. Use the county table above. What was the percent of decrease in Bell County’s population from 2000 to 2010?
Explanation: Only Bell dropped: 21,000 − 19,950 = 1,050. 1,050 ÷ 21,000 (the original) = 0.05 = 5%.
| County | 2000 pop. | 2010 pop. | Area (sq mi) |
|---|---|---|---|
| Adams | 8,400 | 10,500 | 700 |
| Bell | 21,000 | 19,950 | 350 |
| Cook | 5,500 | 5,830 | 1,100 |
| Davis | 60,000 | 66,000 | 440 |
2. In the library pictograph, each symbol is 40 books. How many books were checked out at Eastside and Westgate combined?
Explanation: Eastside has 3 symbols and Westgate has 7, so 3 + 7 = 10 symbols, and 10 × 40 = 400 books.
3. How many more books were checked out at Downtown than at Eastside?
Explanation: Downtown has 5.5 symbols and Eastside has 3, a difference of 2.5 symbols. 2.5 × 40 = 100 books.
4. How many books were checked out at all three branches combined?
Explanation: The total is 5.5 + 3 + 7 = 15.5 symbols, and 15.5 × 40 = 620 books.
5. Using the county table above, how many people per square mile lived in Davis County in 2010?
Explanation: Density = population ÷ area = 66,000 ÷ 440 = 150 people per square mile.
| County | 2000 pop. | 2010 pop. | Area (sq mi) |
|---|---|---|---|
| Adams | 8,400 | 10,500 | 700 |
| Bell | 21,000 | 19,950 | 350 |
| Cook | 5,500 | 5,830 | 1,100 |
| Davis | 60,000 | 66,000 | 440 |
6. A parking lot pictograph uses a car symbol that stands for 30 cars. A row shows 6 1/2 symbols. How many cars does the row represent?
Explanation: 6.5 × 30 = 195 cars.
2. Bar Graphs and Double-Bar Graphs
A bar graph uses the length of bars to show amounts. One axis has a number scale and the other has labels. To find a value, follow the top of the bar across to the scale; if it falls between marks, estimate using a number about halfway between. A double-bar graph pairs two bars for each label so you can compare two data sets, and its key says which shade is which. Bar graph questions usually ask you to read two values and then subtract (how many more), add (how many in all), divide (what percent or ratio), or compare.
Worked Example: From the library graph, how many more visits were there in May than in April? From the film graph, Film X’s cost is what percent of its receipts?
May is 34 and April is 18, so 34 − 18 = 16 more (hundreds of visits, that is 1,600).
Film X cost $45 million and earned $60 million. 45 ÷ 60 = 0.75 = 75%.
1. Using the library bar graph, how many more visits were there in June than in April?
Explanation: June is 38 and April is 18, so the difference is 38 − 18 = 20.
2. Between which two consecutive months did library visits drop the most?
Explanation: February to March fell 4, and March to April fell 8. No other change is a decrease, so March to April.
3. To the nearest percent, by what percent did visits increase from April to May?
Explanation: Change = 34 − 18 = 16. 16 ÷ 18 (the original April value) ≈ 0.889 ≈ 89%.
4. What was the total number of library visits (in hundreds) for January, February, and March?
Explanation: 22 + 30 + 26 = 78.
5. Using the film graph, which film made the greatest profit? (profit = receipts − cost)
Explanation: W: 50 − 20 = 30. X: 60 − 45 = 15. Y: 28 − 35 = −7 (a loss). Z: 90 − 50 = 40. The greatest is Film Z.
6. What is the ratio of Film W’s cost to its box-office receipts in lowest terms?
Explanation: Cost to receipts is 20:50, which reduces to 2:5.
3. Line Graphs
A line graph connects points to show how a value changes over time. One axis shows regular time intervals (years, months, weeks) and the other shows a scale. Rising segments mean an increase, falling segments mean a decrease, flat segments mean no change, and a steeper segment means a bigger change. If the graph has two lines, a key tells which is which, and the place where the lines cross is where the two values are equal. Be careful: the steepest segment has the greatest amount of change, but it does not always have the greatest percent of change, because a small starting value can make a small change a large percent.
Worked Example: In the graph, what was the last year in which Park B’s ticket cost more than Park A’s?
Follow the lines: B is higher in 2015, 2016, and 2017 (26 > 22). In 2018 Park A is $28 and B is $27, so A has passed B and stays above it. The last year B was greater is 2017.
1. In the line graph above, what is the last year Park B charged more than Park A?
Explanation: B is $26 and A is $22 in 2017. In 2018, A ($28) is higher, and it stays higher.
2. How much more did a ticket to Park A cost than a ticket to Park B in 2020?
Explanation: Park A is $38 and Park B is $29, so the difference is $9.
3. In which year-to-year interval did Park A’s price rise the most?
Explanation: The changes are 3, 4, 6, 5, and 5. The greatest, 6, is from 2017 to 2018.
4. To the nearest percent, what was the percent of increase in Park B’s ticket price from 2015 to 2020?
Explanation: Change = 29 − 24 = 5. 5 ÷ 24 ≈ 0.208 ≈ 21%.
5. To the nearest percent, what was the percent of increase in Park A’s price from 2016 to 2018?
Explanation: Change = 28 − 18 = 10. 10 ÷ 18 (the 2016 original) ≈ 0.556 ≈ 56%.
4. Circle Graphs
A circle graph (or pie chart) shows how a whole is split into parts. The whole circle is 100%, so the sections must add to 100% (or to a total amount such as one dollar). Sections can be labeled with percents, fractions, decimals, or amounts, and the size of a section matches its share of the circle (a 25% section is one quarter of the circle). To find the amount for a section, multiply the total by the section’s percent (as a decimal). To find a percent for several sections, add them. In a graph labeled in cents, such as “how a dollar is spent,” 1 cent is 1%.
Worked Example: A household budget is $3,200 a month. (a) How much goes to food? (b) What percent goes to food and transportation together?
(a) Food is 25% of $3,200: 3,200 × 0.25 = $800.
(b) 25% + 15% = 40%.
1. Using the budget circle graph, how much is spent on food each month?
Explanation: 25% of 3,200 is 3,200 × 0.25 = $800.
2. What fraction of the budget goes to transportation, in lowest terms?
Explanation: 15% = 15/100 = 3/20.
3. What percent of the budget goes to rent and savings combined?
Explanation: 40% + 12% = 52%.
4. How much of the monthly budget is saved?
Explanation: 12% of 3,200 is 3,200 × 0.12 = $384.
5. Which section is exactly one quarter of the circle?
Explanation: One quarter is 25%, which is the food section.
6. A circle graph shows how a company spends each dollar: 45 cents on materials, 30 cents on labor, 15 cents on shipping, and the rest on other costs. What percent of each dollar is spent on items other than materials?
Explanation: Cents are percents of a dollar, so everything except materials is 100 − 45 = 55%.
5. Frequency Tables and Tally Marks
A frequency table shows how often each item or value occurs in a data set. Tally marks are a quick way to count: each vertical stroke is 1, and a group of five is shown by four strokes with a slash through them (written here as groups of |||| separated by spaces). The count in a row is the frequency. When data are numbers, they are often put in intervals such as 18 to 24, 25 to 40, and so on; this is a grouped frequency table. To use a frequency table, convert each row to a number, add them to get the total when needed, and then apply ratios, percents, or fractions. If several questions use the same table, write the counts and the total down once so you can reuse them.
Worked Example: A car dealer recorded the colors of the cars chosen for test drives. What is the ratio of black to silver, and what percent of the cars were red?
| Color | Tally | Count |
|---|---|---|
| white | |||| |||| |||| |||| |||| ||| | 28 |
| black | |||| |||| |||| |||| | 20 |
| red | |||| |||| |||| | 15 |
| green | |||| |||| | | 11 |
| silver | |||| |||| | 9 |
| other | |||| |||| |||| || | 17 |
The total is 28 + 20 + 15 + 11 + 9 + 17 = 100. Black to silver is 20 to 9, which is already in lowest terms (20 to 9). Red is 15 out of 100, so 15%.
1. Using the car-color table above, how many more white cars than green cars were chosen?
Explanation: White is 28 and green is 11, so 28 − 11 = 17.
2. What is the ratio of black cars to silver cars?
Explanation: Black is 20 and silver is 9, in that order. They have no common factor, so the ratio is 20 to 9.
3. What fraction of the cars tested were white or black, in lowest terms?
Explanation: 28 + 20 = 48 out of 100, so 48/100 = 12/25.
4. A diner records customers by age group: under 18 has 9, 18–24 has 21, 25–40 has 35, 41–55 has 15, and over 55 has 10. To the nearest percent, what percent of customers were from 25 to 40 years old?
Explanation: The total is 9 + 21 + 35 + 15 + 10 = 90. 35 ÷ 90 ≈ 0.389 ≈ 39%.
5. Using the diner data above (9, 21, 35, 15, 10), what fraction of the customers are 41 or older, in lowest terms?
Explanation: 15 + 10 = 25 out of 90, so 25/90 = 5/18.
6. A returns table shows wrong size 16, defective 3, unwanted gift 20, and changed mind 5. Approximately what percent of returns were an unwanted gift?
Explanation: The total is 16 + 3 + 20 + 5 = 44, and 20 ÷ 44 ≈ 0.45 = 45%.
6. Mean, Median, and Mode
A measure of central tendency (an average) describes the middle or typical value of a data set. The mean is the arithmetic average: add the values and divide by how many values there are. The median is the middle value when the data are in order from least to greatest; with an even number of values, the median is the mean of the two middle values. The mode is the value that occurs most often; a set can have more than one mode, or none if no value repeats. The mean can be a number that is not in the data set, while the median and mode usually are. On a calculator, add the numbers first, press enter, then divide.
Worked Example: Scores in 7 games are 12, 18, 7, 15, 13, 18, and 9. Find the mean, median, and mode. Then find the median of 14, 9, 21, 7, 16, and 11.
Mean: 12 + 18 + 7 + 15 + 13 + 18 + 9 = 92, and 92 ÷ 7 ≈ 13.14.
Median: order them 7, 9, 12, 13, 15, 18, 18. The middle (4th) value is 13.
Mode: 18 appears twice, so the mode is 18.
Even set: order 7, 9, 11, 14, 16, 21. The two middle values are 11 and 14, so the median is (11 + 14) ÷ 2 = 12.5.
1. What is the mean of 8, 12, 9, 15, and 6?
Explanation: The sum is 8 + 12 + 9 + 15 + 6 = 50, and 50 ÷ 5 = 10.
2. What is the median of 14, 9, 21, 7, 16, and 11?
Explanation: Order: 7, 9, 11, 14, 16, 21. With six values, the median is the mean of 11 and 14, which is 12.5.
3. What is the mode of 5, 3, 5, 8, 3, 5, and 9?
Explanation: 5 appears three times, more than any other value.
4. Four weekly paychecks were $412, $380, $455, and $401. What is the mean paycheck?
Explanation: The sum is 1,648, and 1,648 ÷ 4 = $412.
5. The mean of five test scores is 84. Four of the scores are 80, 90, 78, and 86. What is the fifth score?
Explanation: The five scores must total 5 × 84 = 420. The four known scores total 334, so the fifth is 420 − 334 = 86.
6. Which expression gives the mean number of patients per clinic if four clinics have 790, 1,150, 662, and 805 patients?
Explanation: The mean is the sum divided by the number of values, and there are 4 clinics.
7. What is the median of 3.5, 2.8, 4.1, 2.2, and 3.9?
Explanation: Order: 2.2, 2.8, 3.5, 3.9, 4.1. The middle value is 3.5.
7. Line Plots
A line plot shows how often each value occurs along a number line. You place an × or a dot above a number for each data value, so a stack of three means that value occurred 3 times. You can build a line plot from a frequency table, or answer questions from one. Look for four things: the value with the highest frequency (tallest stack, which is the mode), the range (highest minus lowest value), any outlier (a value far from the rest of the data), and the shape of the distribution (clustered, spread out, or uneven). To find the mean or median, first list the values in order using the stack sizes.
Worked Example: The line plot shows how many books 14 students read in a month. Which value is most common, what is the range, and is there an outlier?
The tallest stack is at 3 books. The values run from 0 to 7, so the range is 7 − 0 = 7. The single × at 7 is far from the other values, so 7 is an outlier.
1. In the line plot above, which number of books has the highest frequency?
Explanation: The tallest stack (5 ×s) is over 3.
2. What is the range of the number of books read?
Explanation: Highest value 7 minus lowest value 0 equals 7.
3. Which value is an outlier?
Explanation: 7 is far from the cluster of values between 0 and 4.
4. How many of the students read at least 3 books?
Explanation: 3 books: 5 students; 4 books: 2 students; 7 books: 1 student. 5 + 2 + 1 = 8.
5. What is the median number of books read by the 14 students?
Explanation: The 14 ordered values are 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 7. The 7th and 8th values are both 3, so the median is 3.
6. A frequency table shows a value of 28 with 2 students, 29 with 3, 30 with 0, and 31 with 1. On a line plot, how many ×s should be above 29?
Explanation: The number of ×s above a value is its frequency, which is 3 for 29.
8. Histograms
A histogram is a bar graph for numerical data that has been grouped into equal intervals. The intervals (groups) go on the horizontal axis, the frequencies go on the vertical axis, and the bars touch because there are no gaps between the groups. A histogram can also show percents instead of counts; then the bars add up to 100%. To answer questions, read the height of the bars you need, then add (for several intervals), subtract, or turn the part into a percent of the total. You can also match a histogram to a frequency table by checking that each bar’s height equals the frequency for its interval, and that the order of the bars is correct.
Worked Example: The histogram shows the commute times (in minutes) of 64 workers. How many workers commute for less than 30 minutes, and what percent of the workers commute 60 minutes or more?
Under 30 minutes means the first three bars: 3 + 5 + 9 = 17 workers. For 60 minutes or more, add the last two bars: 6 + 4 = 10. Then 10 ÷ 64 ≈ 0.156 ≈ 16%.
1. In the histogram above, which interval has the greatest frequency?
Explanation: The tallest bar, 15, is the 40–49 minute interval.
2. How many workers commute for less than 30 minutes?
Explanation: 3 + 5 + 9 = 17.
3. To the nearest percent, what percent of the 64 workers commute 60 minutes or longer?
Explanation: 6 + 4 = 10 workers. 10 ÷ 64 ≈ 0.156 ≈ 16%.
4. How many workers are represented in the histogram in all?
Explanation: Add all the bars: 3 + 5 + 9 + 12 + 15 + 10 + 6 + 4 = 64.
5. A frequency table shows 15 scores in 200–399, 22 in 400–599, 35 in 600–799, and 62 in 800–999. Which feature must the matching histogram have?
Explanation: A histogram matches a frequency table when each interval’s bar height equals its frequency, in order from least to greatest interval.
6. A histogram of weekly appointments is changed so that each bar shows the percent of all appointments. If the first two bars are about 2% and 5%, about what percent of appointments occurred in those two weeks?
Explanation: Add the bar values: 2% + 5% = 7%.
9. Simple Probability: Theoretical and Experimental
Probability measures how likely an event is. It can be written as a ratio, fraction, decimal, or percent, and it is always between 0 (impossible) and 1 (certain). Theoretical probability is the ratio of favorable outcomes (what you want) to possible outcomes (everything that can happen) when all outcomes are equally likely. Experimental probability is based on what actually happened: favorable outcomes divided by the number of trials. The two are usually close but not equal. The probability that an event does not happen is 1 minus the probability that it does. Check the answer choices first to see whether they are fractions, decimals, or percents.
Worked Example: (a) On the spinner, what is the probability of spinning a 4? (b) In 20 spins, a 4 came up 6 times. What is the experimental probability?
(a) Two of the 8 equal sections are labeled 4, so the probability is 2/8 = 1/4 = 0.25 = 25%.
(b) Experimental probability = 6/20 = 3/10 = 0.3 = 30%.
1. A game has 50 wooden tiles, and 20 of them are vowels. What is the probability of drawing a vowel?
Explanation: 20/50 = 2/5 = 0.4 = 40%.
2. A bag has 4 red, 4 blue, and 2 green marbles. If one marble is chosen at random, what is the probability that it is green?
Explanation: There are 10 marbles and 2 are green: 2/10 = 1/5.
3. Erin flipped a coin 40 times and got 22 heads. Based on her data, what is the experimental probability of getting tails?
Explanation: Tails came up 40 − 22 = 18 times. 18/40 = 9/20, or 9 out of 20.
4. A deck of 12 cards includes 3 clubs. If a card is chosen at random, what is the probability of selecting something other than a club?
Explanation: There are 12 − 3 = 9 cards that are not clubs. 9/12 = 3/4.
5. A movie theater sold 180 adult tickets and 60 children’s tickets. One ticket is chosen at random for a prize. What is the probability that the winner holds a child’s ticket?
Explanation: The total is 240 tickets, and 60/240 = 1/4. (60/180 compares children to adults, not to all tickets.)
6. A spinner has six equal sections numbered 1 to 6. What is the probability of spinning a 5 or a 6?
Explanation: Two of the six sections are favorable: 2/6 = 1/3.
10. Independent and Dependent Events
To find the probability of two or more events happening, find the probability of each event and multiply. Events are independent when one does not change the other (flipping coins, rolling dice, or drawing an object and then putting it back, which is called replacement). Events are dependent when the first event changes what is possible for the second, such as drawing two marbles without putting the first one back; in that case, adjust the counts for the second event. You can also list the outcomes in a tree diagram. Think carefully about whether an object is replaced before choosing a method.
Worked Example: (a) Two coins are tossed. What is the probability both land heads? (b) A box has 4 blue and 2 red marbles. Two marbles are drawn without replacement. What is the probability that both are blue?
(a) Independent events: 1/2 × 1/2 = 1/4.
(b) First blue: 4/6 = 2/3. If the first marble was blue, there are now 5 marbles and 3 are blue, so the second is 3/5. Multiply: 2/3 × 3/5 = 6/15 = 2/5.
1. Kim rolls two standard six-sided dice. What is the probability that both are 4s?
Explanation: Each die shows 4 with probability 1/6, so 1/6 × 1/6 = 1/36.
2. Ten cards numbered 1 to 10 are in a bag. Toni draws a card, replaces it, and draws again. What is the probability that both cards are greater than 5?
Explanation: Five of the ten cards (6, 7, 8, 9, 10) are greater than 5, so 5/10 = 1/2 each time. 1/2 × 1/2 = 1/4.
3. A spinner has four equal sections: two red, one green, and one blue. If it is spun three times, what is the probability that all three spins are red?
Explanation: P(red) = 2/4 = 1/2 each spin, so 1/2 × 1/2 × 1/2 = 1/8.
4. Twenty marbles are in a bag: ten red and ten black. One is drawn and set aside, then a second is drawn. What is the probability that both are red?
Explanation: First red: 10/20 = 1/2. Second red (one red is gone, 19 marbles left): 9/19. 1/2 × 9/19 = 9/38.
5. There are 15 chips in a bag: 8 green and 7 white. Five white chips are removed. What is the probability that the next chip drawn is green?
Explanation: After removing five white chips, there are 8 green and 2 white, or 10 chips. 8/10 = 80%.
6. In a game, a player rolls a die numbered 1 to 6 and then spins the spinner from the earlier figure (8 equal sections: 1, 4, 2, 4, 3, 5, 2, 6). What is the probability of rolling a 3 and then spinning a 4?
Explanation: P(3 on the die) = 1/6 and P(4 on the spinner) = 2/8 = 1/4. Multiply: 1/6 × 1/4 = 1/24.
11. Combinations (Order Does Not Matter)
A combination is a group chosen from a larger set when order does not matter (apples and bananas is the same group as bananas and apples). For small numbers, count combinations with an organized list or a table. Start with the first item and pair it with each of the others, then move to the second item and pair it only with items after it (to avoid repeats), and so on. For example, with four fruits A, B, G, P and groups of two: AB, AG, AP, BG, BP, GP, which is 6 groups. A helpful shortcut: choosing 4 from 5 is the same as choosing the 1 to leave out, so it has 5 combinations. If the problem says some choices are already fixed, only count the choices that are still open.
Worked Example: Jessica wants to take three of her five books on vacation. How many different groups of three books can she choose?
Label the books A, B, C, D, E and list groups without repeats: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. Counting gives 10 combinations. (Choosing 3 to take is the same as choosing the 2 to leave home, and there are 10 ways to pick two of five.)
1. A supervisor must choose four of five employees for a project team. How many different teams are possible?
Explanation: Choosing which one employee to leave out gives 5 teams.
2. Jessica has five books and will take three. How many combinations of three books are possible?
Explanation: The organized list above gives 10 groups of three.
3. A pizza maker has anchovies, ham, mushrooms, and sausage and will choose three toppings. How many combinations of toppings are possible?
Explanation: Choosing the one topping to leave out gives 4 combinations.
4. Six people are in a club and three will be chosen for a committee (order does not matter). How many committees are possible?
Explanation: List or count: choosing 3 of 6 gives 20 groups. (120 would count different orders as different outcomes.)
5. A chef has 6 vegetables and chooses 2 for a soup. How many combinations of two vegetables are possible?
Explanation: Pair each vegetable with the ones after it: 5 + 4 + 3 + 2 + 1 = 15.
6. Regan will attend a trade show and will take 2 of her 5 employees with her. How many combinations of employees could she choose?
Explanation: Choose 2 from 5: 4 + 3 + 2 + 1 = 10 combinations.
12. Counting Principle and Tree Diagrams
When you must choose one item from each of several different groups, use a tree diagram or the fundamental counting principle: multiply the number of choices for each group. A tree diagram draws a branch for every option at each step, and the number of branch ends is the number of combinations. The multiplication shortcut gives the same answer and is faster when the numbers are large. For example, 3 appetizers × 6 entrees × 4 desserts = 72 meals. Do not add the numbers of choices, and note whether a problem has one big group to choose from (use a list) or several separate groups (multiply).
Worked Example: Sarah has 3 shirts, 2 pairs of pants, and 2 pairs of shoes. How many outfits can she make? Then count the outfits of 2 shirts and 3 pants shown in the tree.
Outfits with all three: 3 × 2 × 2 = 12. For the tree, 2 shirts × 3 pants = 6 outfits.
1. Frank makes a sandwich with 1 of 4 breads, 1 of 3 meats, and 1 of 4 condiments. How many different sandwiches can he make?
Explanation: 4 × 3 × 4 = 48.
2. A restaurant offers 3 appetizers, 6 entrees, and 4 desserts. How many different three-course meals are possible?
Explanation: 3 × 6 × 4 = 72.
3. A passphrase has a Greek letter (24 choices), then a number from 1 to 9, then another Greek letter (24 choices). How many passphrases are possible?
Explanation: 24 × 9 × 24 = 5,184.
4. A doctor will prescribe one of 5 medications, one of 5 dietary changes, and one of 5 vitamin types. How many treatment plans are possible?
Explanation: 5 × 5 × 5 = 125.
5. Julio will choose the beach or a street festival on Friday, one of 3 museums on Saturday, and one of 4 restaurants on Sunday. In how many ways can he plan the weekend?
Explanation: 2 × 3 × 4 = 24.
6. Which of these is a counting problem that uses the fundamental counting principle (multiplying)?
Explanation: The fundamental counting principle applies when one item is chosen from each of several separate groups.
13. Permutations (Order Matters)
A permutation is an arrangement of items in which order matters: coffee-then-museum is different from museum-then-coffee. To arrange all the items in a list, multiply the number of choices for each position, counting down: 4 activities can be ordered in 4 × 3 × 2 × 1 = 24 ways. To arrange only some of the items (for example, first, second, and third place among 10 runners), multiply only as many numbers as there are positions: 10 × 9 × 8 = 720. This is sometimes written P(10, 3). Ask yourself, “Am I finding groups (combinations) or sequences (permutations)?” If a problem uses words such as order, sequence, arrangement, or specific roles, use permutations.
Worked Example: Eight swimmers compete and medals go to first, second, and third place. How many ways can the three medals be awarded? How many ways can a group of 3 swimmers be chosen from the 8 for a relay, if order does not matter?
Medals (order matters): 8 × 7 × 6 = 336.
Relay group (order does not matter): each group of three can be arranged 3 × 2 × 1 = 6 ways, so 336 ÷ 6 = 56 groups.
1. Noemi remembers that her password uses the characters M, Q, $, L, and 7 but not the order. How many possible sequences of the five characters are there?
Explanation: 5 × 4 × 3 × 2 × 1 = 120.
2. A curator must arrange 6 paintings in a line along a wall. How many sequences are possible?
Explanation: 6 × 5 × 4 × 3 × 2 × 1 = 720.
3. Ten runners compete and prizes are awarded for first, second, and third place. How many different results are possible?
Explanation: 10 × 9 × 8 = 720.
4. Ten people want to be extras in a film, and four specific roles (Bystander #1 through #4) will be filled by four of them. How many ways can the roles be filled?
Explanation: 10 × 9 × 8 × 7 = 5,040, because each role is different.
5. Clark’s daughter may choose 3 of 6 rides. How many orderings of three of the six rides are possible?
Explanation: 6 × 5 × 4 = 120.
6. Using the same six rides, how many combinations of 3 rides (order ignored) are possible?
Explanation: Each group of 3 rides appears in 3 × 2 × 1 = 6 orderings, so 120 ÷ 6 = 20.
14. Comparing Proportions in Different Formats
Data can be shown in different ways, such as a decimal, a fraction, a percent, a table, a graph, or a diagram, and the same proportion keeps its size no matter how it is written. To compare two values fairly, convert them to the same format, and a numerical format (decimals or percents) is usually the easiest. Typical steps: read the value from a graph or table, write it as a ratio or rate (the slope of a line, rise ÷ run, gives a rate such as miles per hour per second), turn it into a decimal or percent, and then compare. When asked how close a ratio is to a target, compare the differences from the target.
Worked Example: A new car accelerates from 0 to 60 mph in 8 seconds. An older car’s graph goes from (0, 0) to (10, 72) on a speed-time graph. Which accelerates faster?
The new car: 60 ÷ 8 = 7.5 mph per second. The older car: slope = rise/run = 72 ÷ 10 = 7.2 mph per second. The new car accelerates faster (7.5 > 7.2).
1. A car goes from 0 to 60 mph in 8 seconds. Another car’s graph passes through (0, 0) and (10, 72). Which statement is correct?
Explanation: The first car has 60/8 = 7.5 mph/s. The second car has slope 72/10 = 7.2 mph/s. 7.5 is greater.
2. A graph of a brake test goes from (0 s, 50 mph) to (3.5 s, 0 mph). The competitors’ brakes slow the same car by 15 mph per second. Which is correct?
Explanation: The test brakes slow the car 50 ÷ 3.5 ≈ 14.29 mph/s, which is less than 15 but within 1 mph/s. They are slightly worse.
3. Which is the greatest: 3/8, 0.4, or 35%?
Explanation: Convert to decimals: 3/8 = 0.375, 0.4, and 35% = 0.35. The greatest is 0.4.
4. A candidate predicted winning 55% of the class vote. A bar graph shows 11 votes for the candidate, 5 for the second candidate, and 4 for the third. Was the prediction correct?
Explanation: The total is 11 + 5 + 4 = 20 votes, and 11/20 = 0.55 = 55%.
5. Store A says 2/5 of its customers use coupons. Store B says 38% of its customers use coupons. Which store has the greater proportion of coupon users?
Explanation: 2/5 = 0.40 = 40%, which is greater than 38%.
6. An ancient theory says the best ratio of face height to face width is 1.6. A face is 11 inches tall and 7 inches wide, and five models have ratios 1.50, 1.65, 1.70, 1.56, and 1.53. Compared to the models, how close is the face to 1.6?
Explanation: The face has 11/7 ≈ 1.571, which is 0.029 from 1.6. The models differ from 1.6 by 0.10, 0.05, 0.10, 0.04, and 0.07. The face is closer to 1.6 than all five models, so it is closer than all of them.
Common Mistakes to Avoid
- Skipping the labels and the key. A symbol may be worth 40, not 1, and a scale may be in thousands or millions.
- Using extra data from a table. Choose only the columns the question asks about.
- Dividing by the new value instead of the original when finding percent of change from a graph or table.
- Estimating a bar’s value from the wrong gridline. Line your eye up with the top of the bar and read straight across to the scale.
- Finding the median without putting the data in order first, or forgetting to average the two middle values when there is an even number of values.
- Using the mean when the question asks for median or mode, and forgetting that the mean can be a number that is not in the data.
- Counting tally marks wrongly or forgetting to add all the rows when the total is needed for a percent or probability.
- Forgetting to change the second probability when an item is not replaced (dependent events), or changing it when the item is replaced.
- Using permutations when order does not matter, or combinations when order matters. Check for words such as order, sequence, or specific roles.
- Adding the number of choices instead of multiplying them in a counting-principle problem.
Frequently Asked Questions
What is the difference between mean, median, and mode?
The mean is the sum divided by the number of values, the median is the middle value of the ordered data, and the mode is the value that occurs most often.
How do I find the median of an even number of values?
Put the values in order, then find the mean of the two middle values. For 7, 9, 11, 14, 16, 21 the middle values are 11 and 14, so the median is 12.5.
When do I add and when do I multiply probabilities?
Multiply to find the probability that two events both happen (and). Use the probability of each event separately, changing the second one if the first item is not replaced.
How do I know if a problem needs combinations or permutations?
Ask whether order matters. Choosing a committee of 3 is a combination (a group); awarding first, second, and third place is a permutation (a sequence).
What is an outlier?
An outlier is a data value that is far from most of the other values, such as one student who read 7 books when everyone else read 0 to 4.
What is the difference between a bar graph and a histogram?
A bar graph compares separate categories and the bars have gaps. A histogram shows how numerical data fall into equal intervals, so its bars touch, and it is read as frequencies of those intervals.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.