Organize Data with Diagrams

Example 1 Allen, Joel, Lisa, and Michelle each have a favorite sport. The sports are basketball, cycling, soccer, and tennis. Lisa’s favorite sport does not use a ball. Joel once made a three-point shot in his favorite sport.

Allen does not have a tennis racquet. Find each person’s favorite sport. Example 1

Think: What do I need to find in this problem, and what information is given to help me determine the answer? I need to find the favorite sport for each person. I know that there are four people and four sports to pair up with the people.

Think: Make a table to help organize the known information.

Basketball MichelleLisaJoel Allen Cycling Soccer Tennis

Since Lisa’s favorite sport does not use a ball, it cannot be basketball, soccer, or tennis, so it must be cycling. Mark those boxes on the table. Since Lisa is the one who likes cycling, you can mark out cycling for all the other people.

Think: To help make the rest of the choices clear, put in the other given information. What other information will help us to determine each person’s favorite sport?

Joel is the basketball player since he made a “three-point shot.” Mark out basketball for all the other people, and mark out all the other sports for Joel.

Think: Find another piece of information that may help you decide who likes which sport.

Allen is the soccer player because he does not have a tennis racquet. Mark out tennis for Allen, and mark out Michelle for soccer. You can see from the table that Michelle must be the one who likes tennis.

Conclusion: Allen prefers soccer, Joel likes basketball, Lisa enjoys cycling, and Michelle favors tennis.

A school contains 79 boys who are in one or more of three activities: 41 are out for football (F), 31 play in a band (B), and 20 are in the school play (P). Only two participate in all three activities. Example 2

One member of the football team is in the play but not in the band. Five members of the football team play in the band. How many are in the play but not in the band or on the football team? Example 2

Think: I can draw three intersecting sets to represent the three activities. I can fill in the categories given in the problem.

FFBB PP

Think: There are 31 band members, and 5 are accounted for. This leaves 26. There are 20 students in the play. Three are accounted for, leaving 17. Place 26 temporarily in the “band only” compartment and 17 temporarily in the “play only” compartment.

FFBB PP

Think: I can add all the numbers. If the total exceeds 79, then the excess is the number of people who are in both the band and the play.

= 84 The excess is 84 – 79 = 5. Subtract 5 from the “band only” and 5 from the “play only” compartments. Place 5 in the “band and play” compartment.

FFBB PP

Think: I can verify that the total in band is 31 and the total in the play is 20. Band: = 31. Play: = 20. The totals check. Twelve boys are in the play but not in the band or on the football team.

A residence hall room had students from four continents, one each from North America, South America, Asia, and Australia. Jill was from south of the equator, and she and Tanya both spoke only English. Example

Anna spoke English and Spanish, and Maya was still learning English as her second language. Where was each girl from? Example

In a class with 15 girls, 7 played volleyball and 5 were on the basketball team. If 5 girls did not participate in either sport, how many played on both teams? Example

Anne, Doug, and Roy each play one of the following instruments in the band: clarinet, flute, and trumpet. The trumpet player is a boy. Roy bought a new reed for his instrument. Which instrument does each person play? Exercise

Chris, Frank, Orin, and Paresh drive cars made by Chevrolet, Ford, Buick, and Plymouth. Neither Chris nor Paresh drives a car that has the same first letter as his name. Frank has always driven a Mustang. Exercise

Orin has never owned a General Motors car. Which person drives which car? (Hint: Chevrolets and Buicks are General Motors cars.) Exercise

Everyone at a youth activity was wearing navy or red. There were a total of ten people wearing navy, eight people wearing red, and three people wearing both navy and red. How many people were at the youth activity? Exercise

Miss Tipton surveyed the twenty-six students in her music class. She determined that seventeen students liked Chopin while five liked both Chopin and Vivaldi. The remaining students only liked Vivaldi. How many of her students liked only Vivaldi? Exercise

The song leader, youth pastor, and pastor at Trinity Bible Church are Mr. Carter, Mr. Peek, and Mr. López. Mr. Carter lives next door to the song leader. The youth pastor is Mr. Carter’s son-in-law. Exercise

Mr. Carter and Mr. Peek were on the debate team together in high school. Which job does each person have? Exercise

The Calvary youth group periodically goes on mission trips. Of the thirty-two young people in the group, sixteen have been to Mexico, fifteen have been to Canada, and seventeen have been to Puerto Rico on mission trips. Exercise

Two went on all three trips. Five went to both Mexico and Canada, eight went to Canada and Puerto Rico, and six went to Mexico and Puerto Rico. How many of the group have not been on one of these mission trips? Exercise

Aiden, Jeffrey, Lori, and Nancy live in the towns of Athens, Jackson, Liberty, and Newark. Aiden has spent all of his life in Jackson. Neither Lori nor Nancy has ever been to Athens. Nancy lives in a town whose name has two syllables. Which person lives in which town? Exercise

Determine each girl’s favorite among three drinks, and note any relationships among the girls. One girl’s name is Sabina, and one of the drinks is cola. Two sisters have the same favorite drink. Exercise

Belinda prefers a drink all the other girls hate. Haley does not like root beer. Ashleigh is one of the two who likes lemonade, and Haley is Ashleigh’s cousin. Exercise