# CTK Insights

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16 Apr

### When Two Wrongs Make One Right

How would you go about solving this word problem:

Jerry is twice as old as John; the sum of their ages is that of Jim's, while the ages of the three of them add up to 36. How old is each?

You are likely to start with algebra: let's the ages of Jerry, John, and Jim be $x, y, z$. Then we have three equations:

$x = 2y$,
$z = x + y$,
$x + y + z = 36$.

Solving the equations is not difficult. But here is a method that depends more on arithmetic than algebra. This is the method that would have been used in the ancient Babylon or Egypt.

Let's make a guess. Since Jerry's age could be expressed in terms of John's, we shall start with guessing John's age. Let's assume that John is $4$. Then Jerry is $8$ and Jim is $12$ $(=4+8)$. The sum of their ages comes to $4 + 8 + 12 = 24$. An undershot; the first "wrong" implied in the caption is $24 - 36 = -12$. Let's then guess again and won't be afraid to get a second guess wrong. Assume that Jerry is $7$ years old. This make John $14$ ($=2\times 7$) and Jim $21$ ($=7+14$). Their ages add up to $7 + 14 + 21=42$. Hm, an overshot. The second "wrong" is $42-36=6$.

So far we got two guesses: $4$ and $7$ that led to two wrong calculations: $-12$ and $6$. Now an ancient text could advise to multiply the first guess by the second wrong; the second guess by the first wrong; and divide the difference by the difference of the wrongs. Formally:

$\mbox{right} = \frac{(\mbox{first guess})(\mbox{second wrong}) - (\mbox{second guess})(\mbox{first wrong})}{(\mbox{second wrong})-(\mbox{first wrong})}$.

Let's see whether this indeed works: the formula gives

$(4\times 6 - 7\times (-12)) / (6 - (-12)) = (24+84)/18=108/18=6$.

So we get a suggestion: guess Jerry's age as $6$. Doing that gives John's age as $12$ and Jim's as $18$, with the sum of $36$, as required.

The method - as described - often goes by its Latin name: Regula Falsi, the Rule of the false position. In modern numerical analysis (which is the branch of mathematics concerned with finding approximate solutions to equations) it goes under the moniker of "The Secant Method". In the next blog I'll tell you why.

#### 4 Responses to “When Two Wrongs Make One Right”

1. 1
Mark Dominus Says:

I did something different that is still in the same spirit as the ancient texts. I guessed that Jim was 1, whereupon John was 2 and Jerry was 3. The sum of the ages is 6, but it should be 36, so I sextupled everyone's ages.

I've always thought it was something of a shame that regula falsi isn't more well-known.

2. 2

Mark,

I think you are lucky your method worked. How would you go about solving the problem if I changed the second condition to "the sum of their ages is that of Jerry's less 3"?

I agree with you entirely. Children are rushed to learn the latest and the most expedient, not acquiring much but missing an opportunity to connect to the roots.

3. 3
Ram Says:

Sorry to nitpick, but I think this statement in the solution has the names confused:

"Since John's age could be expressed in terms of Jerry's, we shall start with guessing Jerry's age. Let's assume that Jerry is 4. Then John is 8 and Jim is 12 (=4+8)."

I think it should be:
"Since Jerry's age could be expressed in terms of John's, we shall start with guessing John's age. Let's assume that John is 4. Then Jim is 8 and Jerry is 12 (=4+8)."

A similar change will be necessary in a downstream statement as well.

4. 4