From: Kenneth

How can the following be solved without using algebra?

10 times some number is 3 times the same number plus 14. Answer: 2.

I have a calculation, but I know it is not what the authors used to determine the answer because the topic of algebra was not yet mentioned in the section of the book. It is as follows: 10x = 3x + 14.

Hi Kenneth. Paul sends us three methods not involving algebra:

Method 1 - trial and error
Make any guess: let's try 20
ten times 20 is 200
three times 20 plus 14 is 74
Not equal so try again - the guess is too big

Let's try 10
ten times 10 is 100
three times 10 plus 14 is 44
Not equal so try again - guess is still too big

Let's try 5
ten times 5 is 50
three times 5 plus 14 is 29
Not equal so try again - guess is still too big

Let's try 1
ten times 1 is 10
three times 1 plus 14 is 17
Not equal so try again - now guess is too small

Let's try 2
ten times 2 is 20
three times 2 plus 14 is 20
Therefore, the answer is 2.

Method 2 - using multiples
Multiples of ten always end with a zero
The multiples of 3 are: 3, 6, 9, 12, 15, ...
We need a multiple of 3 such that when 14 is added to it, the answer ends with zero (is a multiple of 10)
The multiples of 3 plus 14 are: 17, 20, 23, ... (the second one in this list has the desired property)
three times 2 plus 14 is 20
ten times 2 is 20
Therefore, the answer is 2.

Method 3 - reasoning with objects
Suppose some object, like a block, represents the answer.
Then, the problem translates to:
10 copies of the block is the same as 3 copies of the block plus 14
Thinking in terms of balancing weight:
10 blocks will balance 3 blocks plus 14
But 10 blocks is the same as 3 blocks plus 7 blocks, so:
3 blocks plus 7 blocks will balance 3 blocks plus 14
By "removing" the three blocks on each side, we are left with:
7 blocks will balance 14
Hence, each block must be 2 (since 2+2+2+2+2+2+2=14).
Therefore, the answer is 2.

Paul Betts