# Solve Problems by Finding Equivalent Ratios

Videos and solutions to help Grade 6 students learn how to solve problems by finding equivalent ratios.

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### New York State Common Core Math Grade 6, Module 1, Lesson 6

Lesson 6 Student Outcomes

Students use tape diagrams to solve problems when given a ratio between two quantities and a change to those quantities that changes the ratio.

Lesson 6 Summary

When solving problems in which a ratio between two quantities changes, it is helpful to draw a ‘before’ tape diagram and an ‘after’ tape diagram.

NYS Math Module 1 Grade 6 Lesson 6 Exercises
Exercise 1:
The Business Direct Hotel caters to people who travel for different types of business trips. On Saturday night there is not a lot of business travel, so the ratio of the number of occupied rooms to the number of unoccupied rooms is 2:5. However, on Sunday night the ratio of the number of occupied rooms to the number of unoccupied rooms is 6:1 due to the number of business people attending a large conference in the area. If the Business Direct Hotel has 432 occupied rooms on Sunday night, how many unoccupied rooms does it have on Saturday night?

Exercise 2:
Peter is trying to work out by completing sit-ups and push-ups in order to gain muscle mass. Originally, Peter was completing five sit-ups for every three push-ups, but then he injured his shoulder. After the injury, Peter completed the same amount of exercises as he did before his injury, but completed seven sit-ups for every one push-up. During a training session after his injury, Peter completed eight push-ups. How many push-ups was Peter completing before his injury?

Exercise 3:
Tom and Rob are brothers who like to make bets about the outcomes of different contests between them. Before the last bet, the ratio of the amount of Tom’s money to the amount of Rob’s money was 4:7. Rob lost the latest competition, and now the ratio of the amount of Tom’s money to the amount of Rob’s money is 8:3. If Rob had \$280 before the last competition, how much does Rob have now that he lost the bet?

Exercise 4:
At the beginning of the year, there was 1 middle school student for every 7 high school students in the select chorus. By the end of the year, for every 3 middle school student in the chorus, there were 5 high school students. If the number of students in the chorus were the same throughout the year, and there were 49 high school students at the beginning of the year, how many middle school students were there in the chorus at the end of the year? NYS Math Module 1 Grade 6 Lesson 6 Problem Set

1. Shelley compared the number of oak trees to the number of maple trees as part of a study about hardwood trees in a woodlot. She counted 9 maple trees to every 5 oak trees. Later in the year there was a bug problem and many trees died. New trees were planted to make sure there were the same number of trees as before the bug problem. The new ratio of the number of maple trees to the number of oak trees is 3:11. After planting new trees, there were 132 oak trees. How many more maple trees were in the woodlot before the bug problem than after the bug problem? Explain.

2. The school band is comprised of middle school students and high school students, but it always has the same maximum capacity. Last year the ratio of the number of middle school students to the number of high school students was 1:8. However, this year the ratio of the number of middle school students to the number of high school students changed to 2:7. If there are 18 middle school students in the band this year, how many fewer high school students are in the band this year compared to last year? Explain.

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