# Algebra In Action

## 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!

##### Level 1Level 2Level 3Level 4Level 5Level 6DescriptionHelpMore Algebra

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.

 1. Charles and Henry together have 49 marbles, and Charles has twice as many as Henry and 4 more. How many marbles has Henry? Working: 2. In an orchard containing 33 trees the number of pear trees is 5 more than three times the number of apple trees. How many pear trees are there? Working: 3. John and Mary gathered 23kg of nuts. John gathered 2kg more than twice as many as Mary. How many kilograms did Mary gather? Working: kg 4. To the double of a number I add 17 and obtain as a result 147. What is the number? Working: 5. To four times a number I add 23 and obtain 95. What is the number? Working: 6. From three times a number I take 25 and obtain 47. What is the number? Working: 7. Find a number which being multiplied by 5 and having 14 added to the product will equal 69. Working: 8. I bought some tea and coffee for £10.39. If I paid for the tea 61 pence more than five times as much as for the coffee, how much did I pay for the coffee? Working: £ 9. Two houses together contain 48 rooms. If the second house has 3 more than twice as many rooms as the first, how many rooms has the larger house? Working: 10. Divide the number 23 into three parts, such that the second is 1 more than the first, and the third is twice the second. What is the first part? Working: 11. 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? Working: 12. Three sisters have 574 CDs between them. Jess has 15 less than Helen, and Tasha has 4 more than Jess. How many does Tasha have? Working: CDs
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This is Algebra In Action level 4. You can also try:
Level 1 Level 2 Level 3 Level 5 Level 6

## 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.

## Transum.org

This web site contains over a thousand free mathematical activities for teachers and pupils. Click here to go to the main page which links to all of the resources available.

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## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

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## Teachers

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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.

[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.]"

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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

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 4

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.

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