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A Choice of Discounts

A Choice of Discounts



Situation

We are ready to purchase two items.  Let A be the cost of the more expensive item, while B is the cost of the less expensive item (A > B).  We have two coupons:

Coupon 1:  Take 10% off of all purchases  (10% off the cost of both A and B)

Coupon 2:  Take 20% off of the most expensive item  (20% off of A, B remains at full price)

The store allows for only one coupon to be used per transaction.  Which coupon gives us the best benefit?  Let’s assume that we are only buying two items.

We will not worry about sales tax in this situation.

Calculating the Costs After Discounts

Using Coupon 1:  Let T1 be the total cost using coupon 1.

T1 = (100% -10%) * (A + B)
T1 = 90% * (A + B)
T1 = 0.9 * (A + B)

Using Coupon 2:  Let T2 be the total cost using coupon 2.

Here, A is more expensive, so:

T2 = (100% -20%) * A + B
T2 = 80% * A + B
T2 = 0.8 * A + B

In summary, the total costs are:

T1 = 0.9 * (A + B)
T2 = 0.8 *A + B

Comparing the Two Coupons

Is there a price point of both A and B where the total costs of the two items is the same?  This occurs when T1 = T2.

T1 = T2

0.9 * (A + B) = 0.8 * A + B
0.9 * A + 0.9 * B = 0.8 * A + B
0.1 * A + 0.9 * B = B
0.1 * A = 0.1 * B
A = B

The costs are the same when the cost of both items are the same.

But we assume that A > B.    So, either T1 will be more or T2 will be more.  To determine this, we will need to perform a subtraction.  Let’s pick T1 – T2.

If T1 -T2 > 0, then T1 costs more and using the second coupon, 20% of the more expensive item,  is preferable.

If T1 – T2 < 0, then T2 costs more and using the first coupon, 10% of both items, is preferrable.

T1 – T2
= 0.9 * A + 0.9 * B – (0.8 * A +  B)
= 0.1 * A – 0.1 * B
= 0.1 * (A – B)

Since A represents the cost of the more expensive item, A > B, and A – B > 0.  Therefore 0.1 * (A – B) > 0 and T1 > 0.   The second coupon is preferable.

Case Studies

A = $29.99,  B = $24.99:  T1 = $49.48,  T2 = $48.98  (not much difference)

A = $29.99,  B = $13.99:  T1 = $39.58,  T2 = $37.98

A = $29.99, B = $5.99;  T1 = $32.38,  T2 = $29.98  (difference between T1 and T2 increase as discrepancy of costs increase)

Hopefully this helps on all of our future shopping trips,

Eddie

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