The tape diagrams from the previous lesson now will be used as a gateway to solving the same type of problems using algebraic equations. Some students may find solving the problems easier using algebra. Others will be more dependent on the tape diagrams, and will use them to derive the algebraic equations. In either case, I view this lesson as a seed. I’ve had students who when they take algebra and have the same type of problems remark that now they seem so much easier than they did in seventh grade.
Warm Ups. The warm-ups include yet another problem about averages—the problems continue to build up in complexity. Other problems are ratio problems that require tape diagrams per the previous lesson.
1. Tim’s average score on the last four math tests is 95. If on his fifth test his average is 96, what is the total of his five scores? What was his score on the fifth test? Answer: Find total for current average: 4 x 95 = 380.
Score on fifth test: Find total for average of 96 with five tests: 5 x 96 = 480. 480-380 = 100, which would have to be the score on the fifth test.
2. Peter saves $52. Susan saves $20 more than Peter. Find the ratio of Peter’s savings to Susan’s savings in simplest terms. Answer: 52:72 = 13/18
3 Express the ratio of 40 minutes to 2 hours in simplest terms. Answer: 40:120 = 1:3
4. The ratio of girls to total number of students in class is 5:12. There are 14 more boys than girls. Find out how many boys and girls there are. Answer: Drawing the tape diagrams, there will be 7 more boxes representing students which would represent the number of boys. Since there are 14 boys, 14 ÷ 7 = 2, and 2 goes in each box. No. of girls = 5 x 2 = 10; No. of boys = 14 + 10 = 24.
5. The ratio of the number of apples to the number of oranges is 7:4. There are 60 oranges. How many apples are there? Answer: Since 4 boxes represents number of oranges and there are 60, then each box in the tape diagram equals 15. No. of oranges = 7 x 15 = 105
Problem 1 may give the students some difficulty. A helpful prompt is to ask what the totals of the scores represent—that is, what does the total of the five scores have that the total of the four scores does not? For Problem 4, the tape diagram will show 7 more boxes representing students. Ask what the additional 7 boxes represents—more girls or boys? Problem 5, which boxes would represent oranges? How many boxes are there?
The Algebraic Approach. I start out by giving a problem and having students draw the tape diagrams.
There are 36 students at a dance and the ratio of girls to boys is 4:5. How many of each are there?
The tape diagrams are drawn in the usual manner.
Students will usually go ahead and solve. I’ve tried telling them not to, but after a few years of students beating me to it, I’ve had to change strategy.
“I can see that most of you have solved this; you put 4 in all the boxes and calculated the number of girls and boys. For girls you multiplied 4 x 4 boxes. For boys you multiplied 4 x 5 boxes. But now I want to do it a little differently. Let’s pretend that we don’t know what goes in the boxes.”
I have to say that in all the time I’ve been doing this, no one has ever said “But we do know!” This isn’t to say that won’t happen.
“When we don’t know what a number is in a problem, what do we use?”
Students will generally say “x”. “OK, I’m going to put x in all the boxes.” I do this on the diagram I have on the board.
“For the number of girls, instead of multiplying a number by 4 boxes, what do we multiply by 4 and how do we write it?” Someone will come up with 4x. (If they don’t, I’ve been known to do my best imitation of Jaime Escalante in the movie “Stand and Deliver” and looking upward say “Forgive them, they know not what they do.” I did this in a Catholic school once. There was no response.)
“We do the same thing for number of boys; how do I represent them algebraically?” I usually hear 5x right away.
“We know that the total number of boys and girls equals what?” Hearing 36, I continue. “If the number of girls is 4x and the number of boys is 5x, and the total is 36, how do I write this as an equation?”
There is usually some hesitancy but someone generally will come up with the answer: 4x + 5x = 36. Combining terms, 9x = 36, x =4.
“We’ve found the number that we normally put in the boxes, but we did it with algebra. But we haven’t solved the problem; we solved for x. But x is the multiplier here. The number of girls is 4x. What does that equal?” They see where I’m going with this and tell me it’s 16. Similarly, the number of boys is 5x which equals 20.
Examples. We now work with examples, which I have them write in their notebooks rather than mini-whiteboards, because I want them to have these for reference.
The first example we do together: “The ratio of roses to tulips is 2:3. There are ten more tulips than roses. How many of each are there?”
I want to know how to represent the number of roses and the number tulips in terms of x. “Draw a tape diagram if you have to, but fill in the boxes with x.” The tape diagram helps them and we soon have: 2x = no. of roses; 3x = no. of tulips, which I have them write down.
“This is a ‘More of one thing than another” type of problem. How do we solve this?”
After I hear a correct answer, I tell them to write this down:
Number of tulips – number of roses = 10;
“You wrote down the number of roses and tulips in terms of x, so let’s substitute them in the sentence to make it an equation that we can solve.”
3x – 2x = 10; x = 10
“Have we solved the problem?”
Some students will say “yes” and some “no”. I want to make clear that we want to know the number of tulips and roses; x is the multiplier. We look back at what was written. “3x is the number of tulips, so what does that equal? What is the number of roses?” We soon have the answer: 30 tulips, and 20 roses.
For the next few, I want them to work it themselves, and I walk around to answer questions and see if they’re on the right track as they write down the problem and steps in their notebooks.
“The ratio of girls to boys at a football game is 3:5. There are 30 boys. How many girls are there?
For this, I provide initial guidance. “Which number, 3x or 5x is going to help us here?” Some students will rely on a tape diagram to do this. “What is the number the problem gives us?” is another prompt I’ll give. I’m trying to lead them to 3x = 30. Solving it yields x = 10, so that the number of girls can now be calculated: 5x = 50.
I usually do a few more, usually picking some that we did the day before using tape diagrams, or modifying them slightly. I then make the following statement: “You may be wondering why we would bother with algebraic equations when we can draw tape diagrams.”
There is usually some agreement here, so I give them the following problem and ask which method they think would be easier:
“The ratio of gallons of oil to gallons of water in a container is 33:78 and totals 333 gallons. How many gallons of each are there?”
The equation is 33x + 78x = 333; 111x = 333, and x = 3. Therefore, there are 33∙3 or 99 gallons of water, and 78∙3 or 234 gallons of oil in the container.
Homework. As mentioned earlier, some students will rely on the tape diagrams to translate the problem into equations. For that reason on the homework worksheet, a sample of which follows, I include problems with larger numbers that discourage the use of tape diagrams.
All that said, some students will avoid writing equations. A seventh grade boy in one of my classes resisted writing equations, or calculations for that matter. He liked solving problems in his head. For problems like the above he would say “I just added the two ratios and then I divided 333 by 111, and then multiplied 3 by 33 and 3 by 78.” And although this is correct and mental math is worth developing, the key is to develop the skill of translating problems into equations. For students such as the one described, I would keep after them. Of course the prospect of getting zero points on a test question because there was no equation would work as an incentive.
Sample Worksheet. Below is a worksheet that I’ve used for homework problems. Some of the problems came from Dolciani’s “Basic Algebra”.
Solving ratio problems using algebra: Worksheet
Use algebra to write an equation and solve. If you have difficulty, use a tape diagram to write the algebra equation.
1. The ratio of roses to tulips is 2:5. There are nine more tulips than roses. How many of each are there?
Answer: 2x = no. of roses; 5x = no. of tulips
Number of tulips – number of roses = 9; 5x-2x = 3x; 3x = 9, x= 3
2x = 6 roses; 5x = 15 tulips
2. In the school band, the ratio of clarinet players to trumpet players is 7:3. There are 30 clarinet and trumpet players altogether. How many of each are there?
Answer: 7x + 3x = 30; 10x = 30; x = 3; 7x = 21 clarinets, 3x = 9 trumpets
3. Peter is 6 times as old as Ella.
a) What is the ratio of Peter’s age to Ella’s age? Answer: 6:1
b). Peter is 15 years older than Ella. How old are Peter and Ella?
Answer: (Peter’s age Ella’s age = 15) 6x-x = 15; 5x = 15; x = 3; Peter’s age: 6x = 18; Ella’s age: x=3
4. Sandy makes bread in which the ratio of whole-wheat to white flour is 3:1. How much white flour is in the total of 8 cups?
Answer: Whole wheat: 3x White: x Equation: 3x+x= 8; 4x=8; x = 2
Whole wheat: 3x = 6 cups White: x = 2 cups
5. A dime contains copper and silver in the ratio of 1:9. How much of each is in 50 pounds of dimes? Answer: 9x+x = 50; 10x = 50; x = 5. Copper: 9x=45 Silver: x=5
6. At Tidewater Tech, the ratio of full-time teachers to part-time teachers is 247:229. There are 18 more full-time teachers than part-time. How many of each are there?
Answer: 247x – 229x = 18; 18x = 18, x = 1. There are 247 full time and 229 part-time teachers.
7. An 80-meter cable is cut into two pieces. The lengths of the pieces have a ratio of 11:5. How long is the shorter piece? Answer: 11x+5x = 80; 16x = 80; x = 5
Shorter piece = 5x = 25 meters
8. There are 35 students at a game. The ratio of boys to girls is 4 to 3. How many boys are there? Answer: 4x+3x = 35; 7x=35; x= 5. Boys: 4x=20, Girls: 3x = 15
9. A marching band has 40 clarinet players. The ratio of members who play clarinets to all others is 2:3. How many non-clarinet players are there in the band? Answer: 2x=40, x = 20. Non-clarinet players: 3x = 60.