From Solving Problems to Investigating Them

 A student was simply solving a routine word problem of Class 10 when he noticed something strange.

A tiny shortcut appeared to give the answer instantly.

The class was impressed.

Then someone asked the question that turns school mathematics into mathematical investigation.

The Original Question:

A man bought some pens for ₹5760. If each pen had cost ₹8 less, he would have received 10 more pens for the same amount.

Find:

1. The original cost of each pen
2. The number of pens bought originally

Most students solved it algebraically.

Let the original cost per pen be p.

Then the original number of pens is: 5760/p

If the price decreases by ₹8, the new price becomes:
p-8

And the new number of pens becomes: 5760/(p-8)

Since he gets 10 more pens:
(5760/p) +10 = 5760/(p-8)

5760(p−8)+10p(p−8)=5760p.

$$5760p-46080+10p^2-80p=5760p.$$

10p^2−80p−46080=0.

p^2−8p−4608=0.

Solving gives: p = 72

• Original cost = ₹72
• Original number of pens = 5760/72 = 80

At this stage, it looked like a typical textbook problem-solving session.

But then something interesting happened.

A Student Notices a Strange Shortcut

One student casually observed:

“The reduction in price is 8 and the increase in quantity is 10.
8 times 10 = 80.
That itself gives the original number of pens!”

The class paused.

It worked.

The original number of pens really was 80.

The shortcut looked magical.

Immediately students felt:

“This is clever.”

And naturally the next question arose:

“Will this always work?”

The Classroom Starts Experimenting

The teacher now asked the class:

“Create another similar question where this shortcut works.”

Students enthusiastically began changing numbers.

Someone tried:

• reduction in price = 3
• increase in quantity = 7

The shortcut predicted:
3 times 7=21

So, students attempted to build a question around:

• 21 items,
• some suitable price,
• some total amount.

One attempt looked promising:

• Original number = 21
• Original price = ₹30
• Total amount = ₹630

But checking carefully:

• New price = ₹27
• New quantity:

630/27 = approx. 23

The increase was not 7.

The shortcut failed.

Another attempt:

• reduction = 4
• increase = 5

The shortcut predicted:
4 times 5=20

Students tried:

• original number = 20
• original price = ₹42
• total = ₹840

Checking:

• New price = ₹38
• New quantity:
  840/38

Again, the increase was not 5.

Failure again.

Some of the students realized with such failures that selection of original price can play such trick that will make the short cut work.

Now the classroom atmosphere changed.

The shortcut was not universally true.

But then why had it worked so perfectly in the original problem?

The teacher wanted the students to investigate deeply (Earth element of PanchTatva learning strategy) and understand if there is any structure hidden in the question that makes the short cut work. If so, can we find such condition using which we can generate more such questions?

At this point the problem stopped being routine algebra and became a mathematical investigation.

Investigating the Structure

Few students tried to find the condition which will make the short cut work.

Suppose:

• Original number of items = n
• Original price = p
• Reduction in price = a
• Increase in quantity = b

Then:
np=(n+b)(p-a), because both are equal to the amount used.

np=(n+b)(p-a)

Expanding:
np=np-an+bp-ab

Simplifying:
an=bp-ab

Now the shortcut claimed:
n=ab

Substituting for n,
a(ab)=bp-ab

a^2b=b(p-a)

a^2 = p - a

p=a(a+1)

So, if this condition is satisfied the shortcut should work.

The Original Problem Revealed Its Secret

In the original question:

• reduction a=8
• increase b=10

So:
p = 8(8+1) = 72; the condition

and

n = ab = 80, the short cut

Exactly the original answer.

Now, teacher asked the students to formulate some more such questions where this shortcut works. It was easy:

Take reduction in price a = 5

so, initial price p has to be = a(a+1) = 5*6 = 30 to satisfy this condition.

Take increase in purchase quantity b = 8

Since we have satisfied the required condition for shortcut to work, n = ab = 40

Initial number purchased = ab = 40 (as per the shortcut)

The amount available = 30*40 = 1200 Rs.

So, when price is reduced by 5 from 30 to 25, the new purchase quantity is 1200/25 = 48 which is 8 more than the initial quantity. 

So, another question where this short cut would work is

A man bought some pens for ₹1200. If each pen had cost ₹5 less, he would have received 8 more pens for the same amount.

Find:

1. The original cost of each pen
2. The number of pens bought originally

What Students Actually Learned

Notice what happened in this lesson.

Students did not merely:

• solve an equation,
• factorize a quadratic,
• or manipulate symbols.

They experienced the actual process of mathematics:

1. Solve a problem
2. Notice a pattern
3. Believe it may always work
4. Test examples
5. Fail repeatedly
6. Become curious
7. Build a general model
8. Derive the hidden condition

And the most educational part was not the successful shortcut.

It was the repeated failure.

Because every failed attempt forced students to ask:

“What exactly is special about the original problem?”

That question transformed algebra from a procedure into an investigation.

And perhaps that is where real mathematics begins.