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! 
This is level 4: find numbers when given information about the sum or difference of their multiples. You can earn a trophy if you get at least 9 questions correct and you do this activity online.
InstructionsTry 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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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.
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'.
The sum of two numbers is 25, and the larger is 3 less than three times the smaller. What are the numbers?
Let \(x\) be the smaller number,
then \(3x  3\) represents the larger number.
$$x + 3x  3 = 25 \\ 4x  3 = 25 \\ 4x = 28 \\ x = 7 \\ 3x  3 = 18$$
The numbers are 7 and 18.
Second Example. Mr Y gave $6 to his three boys. To the second he gave 25 cents more than to the third, and to the first three times as much as to the second. How much did each receive?
Let \(x\) be the number of cents the third boy received,
then \(x + 25\) is the number of cents the second boy received,
and \(3x + 75\) is the number of cents the first boy received.
$$x + x + 25 + 3x + 75 = 600 \\ 5x + 100 = 600 \\ 5x = 500 \\ x = 100 \\ x + 25 = 125 \\ 3x + 75 = 375$$
1st boy received $3.75, 2nd boy received $1.25 and the 3rd boy received $1.00.
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 doubleclick the 'Check' button to make it float at the bottom of your screen.
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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.]"