If D equals Diophantus's age and S is the age of his son you will get the following expressions:

These numbers come from the riddle:

5 years married without child

4 years after his son died till he died

Now doing the parenthesis first:

Find the common denominator witch is 84 we get

(14/84)+(7/84)+(12/84)=(33/84)

(33/84) D+S+9

Replace S with (1/2)D and since we know that (42/84) is the same as (1/2) we can replace that as well

(33/84)D+(42/84)D+9=D

(75/84)D+9=D

Move that D's to the same side,

9=(9/84)D

84=D

Even though it isn't asked for we can now find the son's age by using 84 in our equation for S.

(1/2)(84)=S=42

So taking what we have just gone through it's time to make our own and since we are all math nerds I will use "messier" fractions. I will being doing this in steps, stick with the steps.

__Step 1:__Pick proportion.

(1/8),(1/20),(1/15)

I found what their common denominator is.

(15/120),(5/120),(8/120)

__Step 2:__Variables and additional data.

S=spending

B=Balance

**Step 3:**Write the problem

You're going shopping with your friends and you find your favorite band is selling concert tickets for 1//8 your total bank balance. Continuing shopping you find the best outfit and shoes for the concert which cost 1/15 and 1/20 respectively of you balance. Once you get home you look and see that you have 18/30 of what you started with plus the $10 you have left from lunch. How much did your bank balance start at?

__Step 4:__Check!

B=(1/8+1/15+1/20)B+S+10

S=(18/30)

1/8-concert tickets

1/15-outfit

1/20-shoes

S-how much left

10-your change

Common denominator.

15/120+5/120+8/120=28/120

B=(28/120)B+S+10

Plug in S.

B=(28/120)B+(18/30)B+10

Common denominator.

(28/120)+(72/120)=97/120

B=(100/120)B+10

(20/120)B=10

Multiply the reciprocal of (20/120) to both sides.

(120/20)(20/120)B=10(120/20)

B=(120/10)

B=60

That is the final step and since we found a real answer that makes sense with the question asked we have a successful problem to make it more we could always ask:

How much were the tickets?

How much was the outfit?

How much were the shoes?

This is how I worked through the Diophantus's riddle but there are many different ways to do it. Whats your's?

This problem makes me wonder what else can be shown to students in better ways. This problem is a very good way of showing how fraction can work and correspond to problems that are strictly fraction heavy. I know that during high school I like many other hated fractions and this type of problem I think would help student make a connection to fraction. We all know that once students have a connection to what they are learning then they will not only learn it easier but have a better chance of remembering what they have learned.

Nice! Explained your algebraic thinking thoroughly, and made it clear through constructing your own.

ReplyDeleteclear, coherent, complete, content +

For consolidated, it could use a bit of a finish. One framework I recommend is thinking about one or more of "what" (summary - not needed here as you were very clear), "so what" (why is this important or what is important about it), or "now what" (given this, what's next to think about). Either of those last two would be interesting to me here.