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13 Equations and inequalities

!e Rhind Papyrus is a famous document that is kept in the British Key words

Museum in London. It was written in Egypt in 1650 "#$. It is a
list of 84 practical problems and their solutions. It shows how the Make sure you learn and
people of Ancient Egypt carried out mathematical calculations. understand these key words:

Some of the problems are easy to solve using algebra. !is technique simultaneous equations

was not known in Egypt. !ey wrote their problems substitute
and solutions in words, not symbols. trial and improvement
Here, for example, is Problem 24. inequality
solution set

A quantity and its one-seventh added together make nineteen.
What is the quantity?

!e Egyptian solution was like this. See if you can
understand it.
1
Start with 8 because 7 and of 7 makes 8.
7
Find how many eights make up 19.
Quantities of 8: 1 A 8 2 A 16
1 A 4 1 A 2 1 A 1
2 4 8
1
19 is made up of 16 and 2 and 1. !is is 2 and

4
1
and ‘lots’ of 8. ! e same number of sevens gives
8
the required quantity.
1
1
So now we must calculate 2 and and ‘lots’ of 7:
4
8
1
1 lot is 2 and and 1
4 8
1
2 lots are 4 and and 1
2
4
4 lots are 8 and 1 and 1
2
1
Add these together to get 7 lots of 2 and and 1
4
8
1
1
Total: 7 lots are 16 and and .
2 8
5

!is is the answer, which you would write as 16 .
8
Here is a modern solution, using algebra.
Call the quantity x and write an equation:
x
x + = 19
7
Multiply both sides by 7: 7x + x = 133
A 8x = 133
A x = 133 = 16 5
8 8
Algebra makes it much easier to solve mathematical problems!
You will learn more about using algebra in this unit.
124 13 Equations and inequalities