I love shopping for my girls nowadays and buying items in bright pastel colors that will brighten the monochromatic pallet of the winter. In the last couple of days we made a few special trips to pick warm boots and pants. At first I purchased a couple of toddler pants in a strip mall for a "buy one, get second one half price" deal:

Two days later I finally organized the baby's closet and realized I need many more warm pants for the day care and home. In a run-by shopping after a movie I bought three more colorful pairs at a different store that had a "buy 2, get 1 free" offer.

Taking all the tags off I realized that all the pants had the same original price. But what purchase was a better deal or were they all the same? Each of my older kids had a different opinion on that. What do you think?

Your thoughts and suggestions are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.

Two days later I finally organized the baby's closet and realized I need many more warm pants for the day care and home. In a run-by shopping after a movie I bought three more colorful pairs at a different store that had a "buy 2, get 1 free" offer.

Taking all the tags off I realized that all the pants had the same original price. But what purchase was a better deal or were they all the same? Each of my older kids had a different opinion on that. What do you think?

Your thoughts and suggestions are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.

## 14 comments:

Let x equal the regular price of a pair of pants and y equal the sale price

At the first store, you paid x + 1/2x, or 3/2x and got 2 pair, so 3/2x = 2y

y = (3/2x)/2 = (3/2x) * (1/2) = 3/4x, or 75% of the original price.

At the second store, you paid x + x, or 2x and got three pair, so 2x = 3y

y = 2/3x, or 66.7% of the original price.

So, the second deal was the better deal!

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I think that the buy 2, get 1 free is a better deal on a per unit scale. Take for instance, pants priced at $10 each. With a buy one, get on 1/2 off deal, the total for 2 pairs would be $15, which equates to $7.50 each. With the buy 2, get 1 free, the total for 3 pairs will be $20, which comes to $6.67 each. So even though the total is higher, the unit price is lower, representing a better value.

the answer to the puzzle is

buy 2, get 1 free" is a better deal

if the pants are 5$

"buy one, get second one half price" deal: you would pay $7.5 for 2 which his $3.75 each

"buy 2, get 1 free": you would pay $10 for 3 so $3.333 each

Gorica via email

If we consider the price per piece of clothing, then the second deal is a better one ($0.67 per piece) vs. the first one ($0.75 per piece).

However, we must also consider the number of pieces bought. For instance, if we buy only two pieces of clothing, we don't actually get to enjoy the free 3rd piece in the second deal.

We know that we can fully enjoy the first deal if the number of pieces bought is a multiple of 2; the second deal if the number of pieces bought is a multiple of 3. We will always be able to make full use of both deals if the number is a multiple of both 2 and 3 - i.e. multiple of 6. For each set of 6 pieces, the first deal charges $4.5 and the second deal charges $4. We are looking for the point where the bulk-of-6s difference ($0.5 per bulk-of-6) is insignificant to buying one piece of clothing without any discount ($1). This occurs after the 12th piece. Hence we can safely conclude that for >12 pieces, the second deal is always cheaper.

By using a trivial Excel spreadsheet to test for each scenario between 1 and 12 pieces, we find that the only time the first deal is cheaper than the second deal is when buying 2 pieces ($1.50 vs. $2). Both deals charge exactly the same for cases of 1, 4, 5, and 8 pieces.

In sum...

1, 4, 5, 8 pieces: Draw

2 pieces: 1st deal cheaper

3, 6, 7, 9, 10, 11, 12 pieces: 2nd deal cheaper

>12 pieces: 2nd deal cheaper

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Actually, I don't blame your kids for having a divverence of opinion. The answer is kind of hard to show simply.

Situation One

=========

Buy one, get the second one 1/2 price

If each item costs x and you don't have a sale, the total cost is x + x = 2x

If on the other hand, you get one of them for 1/2 price, then your total is

1x + 1/2 x = 1.5x

The % paid is (1.5x/2x) * 100% = 75%

Situation Two

==========

If you buy 3 pairs at the regular price, the cost = x + x + x = 3x

If you buy 2 pairs and get 1 free then the total cost is x + x = 2x

The % is 2x/3x * 100% = 66 2/3 %

Situation two is the better deal.

Say you wanted 6 pair of pants and for simplicity of calculation each were originally priced at $1.

Using the first deal of "buy one get 2nd one half price" you'd have to spend $4.50 while using the second deal of "buy two get one free" you'd have to spend $4.00. In this case the second deal seems to be better.

I'd go for the second deal!

Lulu

There is a better reason why your kids might disagree other than calculations. Suppose you really only want 2 items. You have no use at all for the third item. Then you are better off taking 1 item and the second 1 for 1/2 price. For the two items you are getting 25% off the total price of 2 items.

If you buy two, you save nothing at all by taking advantage of the sale, because you do not want the third item. Whether there is a sale or not, you are paying what you would for 2 items.

Buy 2, get 1 free is a better deal. Say the pants are $10 each. Buy 1, get 1 half off will be $15 for 2 pairs of pants, or $7.50 each. Buy 2, get 1 free will be $20 for 3 pairs of pants, or $6.67 each.

Darn!!!!

Four

====

I hate it when I don't think a problem out.

The next consideration is 4. Is it better to buy 2 getting 1/2 price on the second [doing this twice] or 3 getting 1 free and paying for an extra 1 at full price?

Turns out it doesn't matter.

2*x + x free pays 2x and then 1 more = 3x

(1x + 1/2x)=3x Same thing.

Five

====

Is it better to buy

2 items and get 1 free + 1 item full price and 1 half price = 2x + 1x + 1/2x = 3 1/2 x

or

2 with 1 at full and the second at a half plus another one at full

2*(x + 1/2x) + x = 4x

The first method is the best way. Finally

Six

===

2 * (2 at full price plus 1 free) = 2*(2x ) + 2 frees = 4x

or

3 * (1 at full and 1 at half price) = 3x + 3*1/2x = 4.5x

The best deal (as it would be for 3 would be two sets at full price and the third one free.

I leave it to whoever reads this to generalize

3n + 1 [same analysis as 4]

3n + 2 [same analysis as 5]

3n [ same analysis as 6]

Is the generalization correct?

Great to see a new blog post! I hope this is a sign that things have settled down again with your job, house, and family.

For this puzzle, assume the price of 1 pair of pants is X (and that the price is same for all pairs).

With "Buy one, get 2nd one half price,' you get 2 pairs of pants for 1.5 X, which means that each pair costs (3/4)X.

With "Buy two, get one free", you get 3 pairs of pants for 2 X, which means that each pair costs (2/3)X.

"Buy two, get one free" is a better deal assuming that you need at least 3 pairs of pants. If you don't need 3 pairs ... and the "deal" has enticed you to buy more than you needed, then it is not such a great deal after all.

An aside: one of the supermarkets near my house often has deals such as "buy1 get 2 free" or "buy 2 get 3 free." With these deals, buying more literally saves you money. For example with "buy 2 get 3 free," the store wants you to get 5 of the items, and if you choose to get 3 or 4, the store charges you for each of them. If you get 5, you only pay for 2.

TracyZ

Welcome Back!!! The Buy 2, Get 1 Free is the better deal. With that deal, each pair of pants costs 2/3 of the full price. With Buy 1, Get 1 at 1/2 Price, each pair of pants cost 3/4 of the original price. Of course this assumes that you actually need 3 pairs of pants. Buying what you don't need is rarely a good deal!

The answer I chose was "Buy two get one free" The other choice "Buy one get the next one for 1/2 price" is not the good of a savings. If hoiwever I wanted 4 items I would choose both choices and get an extra one for .50 cents.

Amazing answers!

A puzzle point to everyone.

The only thing I have to add is that intuitively when you buy one and have second for 50% off you are actually paying for the second half price. When you buy two and get the third off, for each of the bought pants you are getting half pants free. Free is always cheaper than any amount.

However, I have to admit that on this specific occasion, the cheaper pants turned out to have loose waist and were falling off my daughter. I had to make a hole in each pair and pull rubber out of the waist part securing it a bit tighter.

Have a great week everyone!

The second deal was clearly better, if you don't need three pair you could always purchase larger sizes for the next year:-) We don't need to assume that the quality of the pants are any different because of the cost....that scenario always varies widely, you as the consumer will judge that at the purchase point.

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