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