Thursday, October 11, 2012

Calculator Tricks - Part 4

Welcome to Part 4 of Calculator Tricks. This is series is where accomplish mathematical tasks with a use of a regular, simple calculator. Part 4 will cover:

* Dealing with Percents
* Shopping
* Simple Interest

Dealing with Percents

With most calculators, if you have to add tax and subtract discounts, you can just execute the operation directly.

However, on some calculators, like the Casio SL-300VC, require a different sequence:
× n % + (to add) and × n % - (to subtract).

Here is how I prefer to work with percents, and it avoids the percent key altogether.

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Add Percent

Let A be the number you want to add N% percent to it. In a shopping application, N represents the sales tax. In construction, N can be thought be allowance for waste.

A + N%
= A + (A × N/100)
= A × (1 + N/100)

Keystrokes: N ÷ 100 + 1 × A
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Example: Add 10% to 19.95.

Keystrokes: 10 ÷ 100 + 1 × 19.95

Result: 21.945

(On most calculators, 19.95 + 10 % works too)
---------------
Subtract Percent

Let A be the number you want to subtract N% from. In terms of shopping, N represents a discount.

A - N%
= A - (A × N/100)
= A × (1 - N/100)

Keystrokes: N ÷ 100 +/- + 1 × A
---------------
Example: Subtract 10% from 19.95.

Keystrokes: 10 ÷ 100 +/- + 1 × 19.95

Result: 17.955

(On most calculators, 19.95 - 10 % works too)
---------------

Shopping

Ever though what your bill would be as you shop? This section will show you what the potential bill will be and hopefully will lead you to make smart shopping decisions, and keep in budget.

Approach:
1. Clear Memory. We will use the memory register to keep track of our purchases.
2. Determine whether the item is subject to sales tax. You can press MR at any time to get a subtotal.
3. Add the total.
4. *If all of your items are subject to sales tax, add sales tax to the total.

If your purchases are "mixed", buying both non-taxable and taxable items (grocery store comes mind):

If the item or service is not subject to sales tax, use the keystroke sequence price M+ .

If the item or service is subject to sales tax, use the keystroke sequence tax factor × price M+ .

where tax factor = (1 + tax%/100)


In California, where I live, generally food, grocery items, and most services are generally not subject to sales tax, while sales of tangible goods are subject to sales tax.


Let's illustrate this strategy by a few examples.
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Example 1: We are at a hardware store and purchasing the following items:

Hammer: $19.99
Nails: $3.95
Staple Gun: $14.95
Measuring Tape: $3.99

All items are subject to 8.5% sales tax.

Well, since we are dealing with just taxable purchases, we can total everything and add sales tax at the end.

Keystrokes:
MC
19.99 M+
3.95 M+
14.95 M+
3.99 M+
8.5 ÷ 100 + 1 = × MR =


Total: $46.52
---------------
Example 2: From an office store, we are purchasing:

Ream of 500 sheets of paper: 3 at $8.99 each
Pack of pencils: $2.99
Pack of pens: $3.95
Calculator: 2 at $9.95 each

All items are subject to 8.5% sales tax. In this case, we can add the sales tax at the end.

Keystrokes:
MC
3 × 8.99 = M+
2.99 M+
3.95 M+
2 × 9.95 = M+
8.5 ÷ 100 + 1 = × MR =


Subtotal: $53.81
With Sales Tax: $58.38
---------------
Example 3: At a grocery store, we are purchasing:

Bag of Grapes: $2.45
Apples: 5 at 99¢ each
Bananas: 6 at 24¢ each
Bread: $4.09 with a 5% discount coupon
Water: 2 gallons at $1.09 each
Box of Ziploc Bags: $3.99 with a 50¢ coupon

Only the Ziploc bag is subject to 8.5% sales tax. Assume coupons take affect immediately (before sales tax). Here we must use the "mixed purchases" strategy. Enter $3.49 for the Ziploc bags.

MC
2.45 M+
(grapes)
5 × 0.99 = M+ (apples)
6 × 0.24 = M+ (bananas)
5 ÷ 100 +/- + 1 = × 4.09 = M+ (bread)
2 × 1.09 = M+ (water)
8.5 ÷ 100 = + 1 = × 3.49 = M+ (Ziploc bags)
MR (final total)

Total bill: $18.69
---------------


Simple Interest

The simple interest formula is:

I = P × R% × T

I = amount of interest
P = amount of the principal
R% = annual interest rate (as expressed as a decimal)
T = time, in years

The total amount paid is principal and interest, in other words, P + I.

---------------
Example 1: A bank makes a short term loan to Fred and Suzy of $1,000. The bank charges 9.6% interest on short term loans. Fred and Suzy have to pay the loan in two months. If Fred and Suzy wait for the two months, how much interest have they paid?

We are looking for I. We have the following:
P = principal = 1000
R% = annual interest rate = 9.6/100
T = time = 2 months * 1 year/12 months

Note have to convert months to years. So the interest paid is:

Keystrokes:
1000 × 9.6 ÷ 100 × 2 ÷ 12 =

The total interest paid is $16.
---------------
Example 2: Terrell is looking over his credit card bill. The balance is $1,540.29. His credit card charges an annual rate of 15.99%. Terrell is planning to make a $300.00 payment. Assuming Terrell does not use his credit card for the next month, what will be his balance?

Variables:
P = $1,540.29 - $300.00 = $1,240.29
R% = 15.99%
T = 1/12 (1/12 of a year)

Keystrokes:
MC (Clear memory)
1540.29 - 300 = M+ (Subtract payment, interest will accrue on $1,240.29.)
MR × 15.99 ÷ 100 ÷ 12 = (Months interest: $16.53)
M+ MR (Add interest to memory)

Terrell's new balance next month would be $1,256.82.
---------------
Example 3: Lita deposited $500 in a Double Your Money CD. The bank will pay her $1,000 when the CD doubles in value. The bank pays 7.5% interest on these deposits. How long will Lita wait?

This time we are looking for T.

Variables:
P = $500
R% = 7.5%
I = $500

Why is I = $500? Lita deposits $500 and will wait for her account to grow $500, to earn $500 in interest. Solving for T, time:

T = I / (P × R%/100) = I × (P × R%/100)^-1 = (P × R%/100)^-1 × I

Keystrokes:
500 × 7.5 ÷ 100 =
÷ ÷ = (37.5^-1)
× 500 = (Display: 13.333333)

So it takes 13 1/3 years to double Lita's CD. She may want to rethink this investment.
---------------

So this wraps up Part 4 of our Calculator Tricks series. Coming up in Part 5 we will tackle two common algebra problems.

Eddie

This blog is property of Edward Shore, 2012.


1 comment:

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