Using a percentage allows us to express this part-to-whole relationship as a whole number instead of as a fraction or decimal; for example “45% of the population” means we are talking about 45 out of every 100 people. In fraction form, this number would be 45/100 and in decimal form it would be 0.45. All three forms tell us the same piece of information.

**Puzzle one**

Diophantus’ was a 3rd Century AD Greek mathematician from Alexandria and sometimes called “the father of algebra”. Little is known of his life but one of his admirers described his life in the following classic algebraic riddle (or a fractional ‘Life Span’ problem).

“Diophantus’ youth lasted 1/6 of his life. He grew a beard after 1/12 more of his life. After 1/7 more of his life, Diophantus married. Five years later he had a son. The son lived exactly 1/2 as long as his father, and Diophantus died just four years after his son’s death. How many years did Diophantus live?”

### Solution

Diophantus’ life span was **84** years.

### Solution explanation

Let ‘X’ equal Diophantus’ total life span in years.

1/6 X + 1/12 X + 1/7 X + 5 + 1/2 X + 4 = X

= (1/6 + 1/12 + 1/7 + 1/2)*X + 9 = X

Using the fractions’ lowest common denominator or 12 * 7 = 84, then:

(14/84 + 7/84 + 12/84 + 42/84)*X + 9 = X & collecting the fractions together gives:

[(14 +7 +12 + 42)/84]*X + 9 = X

= 75/84*X + 9 = X

= 75/84*X – X = -9

Changing the signs on both sides of the equation gives:

– 75/84*X + 84/84*X = 9 or + 84/84*X – 75/84*X = 9

= 9/84*X = 9

Multiplying both sides of = by the fraction 9/48 inversed (48/9) gives:

(9/84 * 84/9)*X = (9 * 84/9) & therefore, the answer is:

X = 84 years

**Puzzle two**

What number am I?

- I am a four digit number.
- My thousands digit is 5 less than 1/6 of the number represented by my hundreds & tens digits.
- My hundreds digit is 1.8 more than 0.025 of the number represented by my tens & units digits.
- My tens digit is 2 less than 125% of my units digit.

### Solution

The number is **3488**.

### Solution explanation

I am a four digit number (ABCD).

- My thousands digit (A) is 5 less than 1/6 of the number represented by my hundreds & tens digits (BC).
**A = 1/6*(BC) – 5** - My hundreds digit (B) is 1.8 more than 0.025 of the number represented by my tens & units digits (CD).
**B = 0.025*(CD) +1.8** - My tens digit (C) is 2 less than 125% of my units digit (D).
**C = 125%*(D) – 2**

What number am I?

Sometimes it is easier to solve a problem from back-to-front or bottom-to-top. Lets’ start with the tens digit (C):

- C = (1.25 * D) – 2 , letting D equal to 0 through 9, we find that D can only be equal to either 4 or 8 in order to obtain a positive whole number for C (3 or 8).
- B = (0.025 * CD) +1.8 and since CD can only equal 34 or 88. Testing for CD = 34: B = (0.025 * 34) + 1.8 = 2.65 (a non-solution). Testing for CD = 88: B = (0.025 * 88) + 1.8 = 4 (a solution)
- A = 1/6 (BC) – 5 and since BC now equals 48, A = (1/6 * 48) – 5 = 3

ABCD = 3488

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