Chapter 21 - Business Arithmetic

This chapter will cover once over lightly the more common JL business expression involving arithmetic.

The first of these is discount, or mark-down. Retail stores figure the discount they get from the manufacturer or wholesaler with the retail price as the base (book stores, hardware stores, most specialized stores) or—just the opposite in some fields— with the net, discounted price as the base (department stores, chain stores, etc.). When the net price is the base, the store figures mark on or mark up, rather than mark down.

The difference becomes clear in a concrete example.

Mark-down

Suppose a lawn mower retailing for $150 comes to the store with a 30% discount. What is the net price to the store?

The base here is $150. Change the percentage to a decimal and simply multiply. The discount in dollars is .30 times $150, or $45. The net price is $150 minus $45, or $105.

Short cut: The quickest way to figure a net price is not to work out the discount in dollars and then subtract, but to mentally convert the discount into its complement (of 100) and multiply the retail price directly by this. If the retailer gets a 30% discount, then he naturally pays 70% of the retail price.

.70 x $150 gives $105 in one operation, without subtracting.

Try one yourself. A typewriter with a list price of $85 carries a 15% discount to the store. What does the retailer pay for it?

The standard way of doing this is to take .15 of $85, or $12.75, and deduct this from $85 to get a net price of $72.25. The short way is to note that the dealer, in getting a 15% discount, pays 85% of the retail price. So we multiply .85 x $85 and, again, get $72.25 in one operation.

Mark-up

The opposite expression used in many fields is to begin with the net price (the discounted price to the dealer) and arrive at a desired selling price by deciding how much mark-up is required.

A store might have a desired 20% mark-up, for instance. If it buys baby carriages at $30 each net, how much should it sell them for?

Mark-up is figured with the net price as the base, rather than the retail price, so 20% of $30 is $6.00. Adding the cost and the mark-up, the store will sell its baby carriages for $36.

Once again, this can be done without adding, in one operation, by considering that adding 20% to the net price is the same as multiplying the net price by 120%. In this case the short cut is not so effective, however, since you add in the process of multiplying anyway.

Work out a proper selling price for an article that costs $47 and should deliver a 40% mark-up to the store.

For this calculation 40% becomes .4, and .4 x $47 is $18.80. Adding $18.80 to $47, we find a desired retail price of $65.80.

Compound Discounts

Frequently discounts from the retail price are quoted in compound or chain fashion. Toy jobbers (local wholesalers who stock toys and resell them to stores) often buy at discounts such as 50% plus 10%, often called "50 and 10."

This discount is by no means as simple as it looks. It is not the sum of 50 and 10; that is, it is not equal to a 60% discount. This is because the second discount is figured on the net price after the first discount, not on the full retail price.

This becomes clear if we start with a $100 item. The 50% discount gives us a first net price of $50. The 10% discount is now applied to the $50, not to the $100, and amounts to $5. This leaves a net-net price of $45. If we had totaled the discounts, we should have figured a net-net price of $40.

The very general 2% cash discount operates in the same way. In order to get their money quickly, most manufacturers allow an extra 2% off the net amount of the bill if it is paid by the 10th of the following month.

If our $100 item came to a jobber on such terms, he could (by prompt payment) deduct 2% of the net price. This is 2% of $45, not of $100, so it amounts to 90¢ rather than $2.00. The 2% is important over the total picture (2% can be the profit-margin in some types of business) even if it does not seem spectacular on this $100 item.

So the net result of buying a $l00-at-retail toy at a discount of 50% plus 10% plus 2% is that you pay $44.10.

It saves time, in a business in which such discounts prevail, to work out equivalents for the most usual combinations. We have just noted that a discount of 50% plus 10% plus 2% is in effect 55.9% off the retail price.

Turn to your pad and, using 100 as a convenient starting point, work out equivalent one-step discounts for the following compound discounts:

30% plus 5%

40% plus 10%

20% plus 10% plus 5%

The equivalent discounts for these three compound or chain discounts are 33%, 46%, and 31.6%. Not nearly as generous as they look—which is the reason for quoting them in compound form. They appear to be better than they really are.

Figuring Discounts

A chair retailing for $26 costs the store $18.20. What is the discount percentage?

This is the familiar problem we covered in the chapter on percentage—the process of finding the percentage of difference. The difference here (subtract net from retail) is $7.80. $7.80 is what per cent of $26?

Remember to divide by the base, the number following "of." Our fraction is 7.80/26:

Move the decimal point two places to the right to convert the decimal to a percentage: 30%.

Suppose we want to know the percentage of mark-up in this same case? The net price is now our base, so the fraction is 7.80/18.20:

We see at once that the next digit of the answer will be 5 or more (2 into 10), so we can move the point over to convert to a percentage and round off to 40.7%.

Note how much more the mark-up is as a percentage than is the discount. This is always true, because the base (the net price) is smaller than the retail price.

Break-even

A common expression in many business endeavors is the phrase "break-even point." There are many special applications, but in general the phrase describes the minimum quantity (or volume) required before a product or operation can break even and begin to make a profit.

In tooling up for a new plastic toy, for instance, a manufacturer may have to spend $20,000 in research and die-making costs. If the toy sells for $1.00 retail and he gives the normal 50% plus 10% discount, then he receives 45¢ for each toy. His selling overhead may be 10%, his raw cost of plastic, manufacture, packing and shipping another 10%, and his general company overhead 20%, or a total of 40% of that 45¢ (since the manufacturer figures his volume on his sales volume, not the retail price). This leaves 60% of that 45¢ to pay back the cost of getting ready to produce the toy, or 27¢ each. How many toys does he have to sell before he begins to make a profit?

The answer is found by dividing the "contribution" of each sale (27¢) into the "plant account," as it is often called:

He will have to sell roughly 68,000 of this toy before he recovers his initial investment. Once that investment has been recovered, however, he stands to make 27 ¢ profit for each toy sold.

The form of the division, by the way, may need a word of explanation. When you start to produce the shorthand remainder of the next 7, you note that the remainder pattern repeats itself. So you know without working it out that the answer will consist of a series of 7's, repeated forever.

A very similar type of calculation is used to determine the break-even point of volume for, say, a grocery store. In any break-even problem, certain assumptions are made about "fixed" costs, such as the plant account for the toy above, or the running expenses of a store, and "variable" costs, or costs that are incurred only when each sale is made.
 
If all the fixed costs for a certain store were $1,000 a month—including rent, salaries, insurance, etc.—and the average net profit before overhead was 12%, then it is not difficult to calculate how much volume this store must do in order to break even. 12% is the contribution of sales to fixed overhead, or 12¢ on the dollar, so we divide the fixed cost again by the contribution:

This store must do over $8,300 a month in sales volume before it can meet its fixed costs. For every dollar above that it does each month, it returns 12¢ profit.

Commission

Salesmen, stockbrokerage houses, insurance agents, and many other companies and people are paid in commission rather than by salary.

Commission is a simple percentage of the gross, or retail price (or net price, depending on the agreement). If a real-estate broker arranges the sale of a house for $20,000 and earns a 5% commission, he gets $1,000.

Commissions vary widely. Salesmen, depending on the field of business, may earn from 1% or 2% to 15% or even more. Stockbrokers work on a sliding scale that goes down as the volume goes up, on the theory that there is about as much paper work in buying or selling $50 worth of stock as there is in buying or selling $100,000 worth. Advertising agencies traditionally get a 15% discount (commission) on the space they buy from magazines or newspapers and the time they buy on radio or television.

As in any percentage situation, you can start with any two known factors and calculate the third, unknown one.

These three cases represent each possible type of unknown. See if you can answer each of them:
 
A salesman is on 6% commission. He makes a $480 sale. How much commission does he earn by this sale?

Another salesman, on 8% commission, earned $64 one afternoon. How much business did he write in order to get the commission of $64?

A third salesman, on orders totaling $1,300, earned $91 in commissions. What is his commission rate?

Cover the answers with your pad as you work these out.

The first salesman merely has to multiply $480 by .06. He earns $28.80.

The second salesman has to find the base. $64 is 8% of what? As you remember from the chapter on percentage, he determines the unknown base by dividing $64 by .08:
 
The third salesman also has to divide, but he divides his commission by the base in order to make sure of his rate. The answer is 7%. Interest
Most of us deal with interest in our personal lives, whether or not we deal very much with it in business. We buy homes almost invariably with a mortgage carrying interest charges. Often we buy automobiles, major appliances, furniture on "time payments" that include interest, whether or not the interest is called that. Sometimes it is called "carrying charges." A bank loan or finance company loan always carries interest charges.

Compound interest is an intriguing subcategory that has little actual utility for most of us. It merely means that the interest is continually added to the principal on which interest is paid, so there is eventually a snowballing effect that can become quite dramatic after a century or two. Except for large interest rates and long periods of time, however, there is little difference in the results.

The interest you receive on your savings account, or the interest you pay on most mortgages, is a "real" interest, figured periodically on the amount of money the bank owes you or you owe the person holding the mortgage.

If you owe $16,000 on a 6% mortgage, the proper charge for one month for the use of this money is 1/12 of .06, or ½ of 1% (.005), which works out to $80. We use 1/12 of the interest rate for one month because interest is (unless otherwise stated) figured by the year.
All fair and square. With your mastery of percentages, you should have no trouble with any problem in this area.

But interest, in today's world, has become quite a different thing for most of us. A lender may "prove" to you in black and white that he is charging you 8 % interest, yet really can be quite legally gouging you to the extent of 16% or even more. This is so important to almost everybody who borrows money any time in his life that it is worth a page of special explanation.

Hidden Interest

Let us show how the most honest, time-honored, and respectable type of loan from the most inexpensive possible place works: a new-car loan from a bank.

Banks are by far the most reliable and safe places with which to do this kind of business. But when you take out a new-car loan and they say you will pay 6% interest, the reality is that you will pay more than 12%. At a finance company, this could easily go over 24% in real interest charges.

This is why:

When you borrow money and agree to pay interest for the use of it, you properly pay interest on the money while you have it. This is the way mortgages and savings accounts work. Each month (or quarter), the interest on the balance owed is figured and you pay it.

But it does not work this way with consumer loans.

Suppose you go to a bank to borrow, say, about $1,100 to help buy a new car. Your credit standing is good, so the bank says, "Fine." They will charge you only 6% interest, deducted in advance. This means that you sign a note for $1,200, payable in 12 monthly installments. From this $1,200 they now deduct 6% interest for the year it will take you to pay back the loan. 6% of $1,200 is $72, so they give you a check for $1,128 and you buy your car.

The real interest on this loan is more than twice the 6% quoted. Why? For two reasons, First, you never got the $1,200 on which you pay the 6%. You got only $1,128. Second, you do not have the money for a year at all. You start paying it back the very next month—but the money you pay back the next month has had interest charged for a full year.

Here is how the interest would be charged if you were paying a real 6% on the amount you owe, making payments every month. We chose a $1,200 note to make the figuring easy, since you pay back $100 a month for a year.

                                                                              Proper Interest
                                                                              For The Month

Month Of Loan            Amount You Owe              At 6% A Year

The real 6% interest on such a loan totals $34.68. The discounted-in-advance-for-the-full-term arrangement has you pay $72—over twice as much.

This illustration is not designed to malign banks. They are the most trustworthy of all such institutions. But if a quoted interest rate at a bank can be so deceptive, imagine what your real charges can become at finance companies when they talk about "only" 8% or 12% a year—discounted in advance.

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