NCERT Class 8 Mathematics Ganita Prakash Part 2 Chapter 06 Algebra Play PDF Download

Chapter 6: Algebra Play

6.1 Algebra Play

Over the last two years, we have used algebra to model different situations. We have learned how to solve algebraic equations and find the values of unknown letter-numbers. Let's now have some fun with algebra. We shall investigate tricks and puzzles, and explain why they work using algebra. We will also see how to invent new tricks and puzzles to entertain others.

6.2 Thinking about 'Think of a Number' Tricks

In Grade 7, we learned about 'Think of a Number' tricks, like this one.

1. Think of a number.
2. Double it.
3. Add four.
4. Divide by two.
5. Subtract the original number you thought of.

I predict you get 2. Am I right? Try it out with different starting numbers. Do you always end up with the same value, 2? Why?

We saw that we can understand such tricks through algebra.

1. Think of a number: \( x \)
2. Double it: \( 2x \)
3. Add four: \( 2x + 4 \)
4. Divide by 2: \( x + 2 \)
5. Subtract the original number you thought of: \( x + 2 - x = 2 \)

Therefore, no matter what the starting number is, the end result will always be 2!

Teacher's Note

When you work through a trick with algebra, always start by giving the unknown number a letter like \( x \). Then follow each step carefully, simplifying the expression as you go. At the end, if all the \( x \) terms cancel out and you are left with just a number, that's your prediction - it will never change, no matter what number someone starts with.

How would you change this game to make the final answer 3? What about 5?

Can you come up with more complicated steps that always lead to the same final value?

Let us now look at a different trick of this type.

[Figure: A flowchart showing a date trick with characters Mukta and Shubham. Mukta thinks of 26/01 and follows steps: Multiply the month by 5 (1 × 5 = 5), Add 6 (5 + 6 = 11), Multiply by 4 (11 × 4 = 44), Add 9 (44 + 9 = 53), Multiply by 5 (53 × 5 = 265), Add the day (265 + 26 = 291). Shubham then says "You thought of our Republic Day, 26/01.", See in your textbook]

How did Shubham figure out the date chosen by Mukta?

• Let the month be \( M \) and the day be \( D \).
• Multiply \( M \) by 5: \( 5M \)
• Add 6: \( 5M + 6 \)
• Multiply by 4: \( 20M + 24 \)
• Add 9: \( 20M + 33 \)
• Multiply by 5: \( 100M + 165 \)
• Add the day: \( 100M + 165 + D \)

Mukta's answer was 291.

• \( 291 = 100M + 165 + D \)
• \( 291 - 165 = 100M + D \)
• \( 126 = 100M + D \)

Since \( D \) is a day within a month, it is atmost 31 and requires only 2 digits. So the last 2 digits are \( D \) and what comes before that is \( M \). In this case, \( M \) is 1 and \( D \) is 26, i.e., the 26th of January.

Mukta thinks of another date, follows the same steps, and reports her answer as 1390. What date did Mukta start with this time?

• Subtracting 165 from 1390, we get 1225.
• This means that the date she thought of was 25th December.

Find the dates if the final answers are the following:

1. 1269
2. 394
3. 296

You can try this trick with your friends. Ask them to choose the starting date as their birthday.

Can you change the steps in this trick and still find the original date? Instead of subtracting 165 from the final answer, you might have to subtract some other number.

Try to devise your own 'Think of a Number' trick.

Teacher's Note

In the date trick, the key is that multiplying by 100 moves the month to the hundreds and tens places. That is why you can read off the month and day separately from the final number. Before you apply this trick with friends, always check: subtract 165 from the answer, then the last two digits give the day and what is left gives the month. If the month does not make sense (like 13 or 0), ask them to try again.

6.3 Number Pyramids

In a number pyramid, each number is the sum of the two numbers directly below it (see the figure).

[Figure: A number pyramid with three rows. Top row: 23. Middle row: 10, 13. Bottom row: 1, 9, 4, See in your textbook]

Key Points

• When you perform a sequence of operations on an unknown number \( x \), you can use algebra to see if the final result is always the same value, regardless of what \( x \) is. If all the \( x \) terms cancel out, you have discovered a trick.
• To reveal a hidden number in a trick, work backwards from the final answer using inverse operations or by solving an algebraic equation.
• The key to the date-finding trick is that multiplying by 100 separates the month and day into different digit positions, making them easy to extract from the final answer.

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