Algebra In Action

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Real life problems adapted from an old Mathematics textbook (A First Book in Algebra, by Wallace C. Boyden 1895) which can be solved using algebra and common sense!

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This is level 5: more questions similar to those in previous levels. You can earn a trophy if you get at least 9 questions correct and you do this activity online.

1. Mary bought some blue ribbon at 7 cents a yard, and three times as much white ribbon at 5 cents a yard, paying $1.10 for the whole. How many yards of the blue ribbon did she buy?	Working:
2. Twice a certain number added to five times the double of that number gives for the sum 36. What is the number?	Working:
3. Mr. James Cobb walked a certain length of time at the rate of 4 miles an hour, and then rode four times as long at the rate of 10 miles an hour, to finish a journey of 88 miles. For how many hours did he ride?	Working:
4. A man bought 3 books and 2 lamps for £14. The price of a lamp was twice that of a book. What was the cost, in pounds, of a book?	Working: £
5. George bought an equal number of apples, oranges, and bananas for £1.08; each apple cost 2 cents, each orange 4 cents, and each banana 3 cents. How many fruit did he buy altogether?	Working:
6. I bought some 2-cent stamps and twice as many 5-cent stamps, paying for the whole £1.44. How many 5-cent stamps did I buy?	Working:
7. I bought 2 pounds of coffee and 1 pound of tea for £1.31; the price of a pound of tea was equal to that of 2 pounds of coffee and 3 cents more. What was the cost, in pence, of one pound of coffee?	Working: pence
8. A lady bought 2 pounds of crackers and 3 pounds of gingersnaps for £1.11. If a pound of gingersnaps cost 7 cents more than a pound of crackers, what was the price, in pence, of a pound of crackers?	Working: pence
9. A man bought 3 lamps and 2 vases for $6. If a vase cost 50 cents less than 2 lamps, what was the price of a lamp in dollars?	Working: $
10. Divide 71 into three parts so that the second part shall be 5 more than four times the first part, and the third part three times the second. What is the smallest part?	Working:
11. Divide the number 288 into three parts, so that the third part shall be twice the second, and the second five times the first. What is the largest part?	Working:
12. Two numbers have a sum of 216 and a difference of 48. What is the smallest of those two numbers?	Working:

This is Algebra In Action level 5. You can also try:
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Instructions Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

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Transum, Sunday, January 26, 2014 "These questions have been adapted from 'A First Book in Algebra' by Wallace Boyden. They all are designed to encourage an algebraic solution by setting up an equation (or alternatively simultaneous equations) and solving it. Some of the questions could be classified under the topic of ratio. In his introduction Wallace Boyden states 'Algebra is so much like arithmetic that all that you know about addition, subtraction, multiplication, and division, the signs that you have been using and the ways of working out problems, will be very useful to you in this study. There are two things the introduction of which really makes all the difference between arithmetic and algebra. One of these is the use of letters to represent numbers, and you will see in the following exercises that this change makes the solution of problems much easier.'." I'm Not A Humanist But..., Planet Earth Saturday, May 24, 2014 "On level 4 of the 'algebra in action' section, question 11 says: "Divide the number 137 into three parts, such that the second is 3 more than the first, and the third five times the second. What is the third part?" The answer is 100, but it was marked as being wrong, so I tried again, but with 20 (the second number) and 17 (the first number) and it marked 17 as being correct. Just thought you should know about this mistake so that you can correct it. [Transum: Thank you so much for taking the time to highlight this error. You were indeed right and the error has now been corrected. Thank you so much.]" Nick Staples, Waipahu Intermediate Wednesday, August 18, 2021 "Is there a way these algebra in action activities can be made into printable worksheets please. [Transum: Thanks for your feedback Nick. Most of the Transum online exercises can be printed directly from your browser. The input boxes transform into dotted lines for pupils to write their answers on and if the page is being printed by a Transum subscriber, the answers appear at the bottom of the worksheet.]" Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

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Description of Levels

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Level 1 - Find two values given their ratio and either their sum or difference

Level 2 - Find one of three numbers given the connection between them

Level 3 - Find numbers whose sum and difference are given

Level 4 - Find numbers when given information about the sum or difference of their multiples

Level 5 - More questions similar to those in previous levels

Level 6 - Yet more questions similar to those in previous levels

Level 7 - The algebra involved in this level includes fractions

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Back in 1895 Mr Boyden wrote 'Algebra is so much like arithmetic that all that you know about addition, subtraction, multiplication, and division, the signs that you have been using and the ways of working out problems, will be very useful to you in this study. There are two things the introduction of which really makes all the difference between arithmetic and algebra. One of these is the use of letters to represent numbers, and you will see in the following exercises that this change makes the solution of problems much easier'.

Example for level 5

Arthur bought some apples and twice as many oranges for 78 cents. The apples cost 3 cents apiece, and the oranges 5 cents apiece. How many of each did he buy?

Let \(x\) be the number of apples
\(2x\) = number of oranges,
\(3x\) = cost of apples,
\(10x\) = cost of oranges.
$$3x + 10x = 78 \\ 13x = 78 \\ x = 6 \\ 2x = 12$$ Arthur bought 6 apples and 12 oranges.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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