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 2: find one of three numbers given the connection between them. You can earn a trophy if you get at least 9 questions correct and you do this activity online.

1. A man bought a hat, a pair of boots, and a tie for £90; the hat cost four times as much as the tie, and the boots cost five times as much as the tie. What was the cost of the tie?	Working: £
2. A woman travelled 90km in three days. If she travelled twice as far the first day as she did the third, and three times as far the second day as the third, how far did she go on the third day?	Working: km
3. James had 30 marbles. He gave a certain number to his sister, twice as many to his brother, and had three times as many left as he gave his sister. How many did he have left?	Working:
4. A farmer bought a horse, cow, and sheep for $900. If he paid three times as much for the cow as for the sheep, and five times as much for the horse as for the sheep, what was the price of the cow?	Working: $
5. Abdul had seven times as many apples, and Betsy three times as many as Colin had. If they all together had 55 apples, how many apples did Betsy have?	Working:
6. A man left £1400 to be distributed among three sons in such a way that James was to receive double what John received, and John double what Henry received. How much did James receive?	Working: £
7. There are 120 pigeons in three flocks. In the second there are three times as many as in the first, and in the third as many as in the first and second combined. How many pigeons in the third flock?	Working:
8. Divide 209 into three parts so that the first part shall be five times the second, and the second three times the third. What is the first part?	Working:
9. Three people, A, B, and C, earned $1100; A earned four times as much as B, and C earned as much as both A and B. How much did A earn?	Working: $
10. Share 770 in the ratio 5:8:9. What is the middle share?.	Working:
11. Share 306 in the ratio 5:6:7 then add the smaller share to the larger share.	Working:
12. A cistern, containing 1800 litres of water, is emptied by three pipes in two hours. Pipe one discharges three times as many litres per hour as pipe two and six times as many as pipe three. How much water does pipe one discharge in an hour?	Working: litres

This is Algebra In Action level 2. You can also try:
Level 1 Level 3 Level 4 Level 5 Level 6 Level 7

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.]"

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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 2

There are three numbers whose sum is 96; the second is three times the first, and the third is four times the first. What are the numbers?

Let \(x\) represent the first number,
\(3x\) represent the second number,
\(4x\) represent the third number.
$$x + 3x + 4x=96 \\ 8x=96 \\ x=12 \\ 3x=36 \\ 4x=48$$
The numbers are 12, 36, and 48.

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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