Speed Mathematics Simplified (64 page)

Note how the decimal point was moved over, following the rule in the chapter on decimals.

90 is 45% of 200.

The reason we developed this method step by step is to emphasize the logical reasoning behind the general rule:

To find an unknown base, convert the percentage to a decimal and divide it into the known number.

Reinforce this rule at once by trying another example. 68 is 20% of what?

Convert the percentage into a decimal and divide it into the known number:

68 is 20% of 340.

Try one by yourself. Cover up the solution with your pad. 87 is 30% of what?

To find the unknown base, convert the percentage into a decimal and divide it into the known number:

87 is 30% of 290.

Percentage of Change

Business arithmetic often involves a percentage of change or difference. Rather than asking what per cent 18 is of 360, the business world is more apt to ask, “How much more is 500 than 475?” or, “How much less is 390 than 415?”

Suppose that sales in territory #8 were $350,000 last
year, and are $375,000 this year. What is the percentage of increase?

The first step is to find the raw amount of the difference in plain numbers. It is $25,000, found by subtracting the total last year from the total this year.

Now our problem is, “$25,000 is what per cent of $350,000?”

This is familiar. You did similar problems a few pages ago. The dollar signs and 0's do not change the principle. In fact, you can simplify matters by dropping both the dollar signs and the
same
number of 0's: 25 is what per cent of 350?

Remember your base, the number following “of.” The fraction is

Work out the division to convert this fraction to decimal form in shorthand division:

The answer is not precise, but we can round it off to 7%. Territory
#
8 is 7% ahead of last year.

The general rule, then, is this:

Find the difference, and divide it by the base.

Sometimes the base is the smaller of the two numbers; sometimes it is the larger. After all, sales in territory #8 might have gone down this year. Then the base would be the larger of the two figures.

Do this one on your pad:

Sales last year $320,000
Sales this year $307,200

What is the percentage of decrease?

When we find a percentage of decrease, our base is the
larger number. The difference in sales, by subtraction, is $12,800. Dividing by the base—dropping thousands and dollar signs—we have.

This territory is, unhappily, 4% behind last year in sales.

Note especially that sometimes you figure the percentage of difference on the smaller of two numbers, and sometimes on the larger. The difference, as a percentage, will be larger when based on the smaller number—and smaller when based on the larger number.

The saving grace, perhaps, is that an increase in sales from $100,000 to $150,000 will show up as a 50% increase, while a decline from $150,000 to $100,000 is only 33%!

Now that we have covered decimals and percentage, we are equipped to cover the more common business expressions such as discount and interest and some of the other yardsticks most frequently used in the commercial world.

21

BUSINESS ARITHMETIC

T
HIS chapter will cover once over lightly the more common 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 × $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 die 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 × $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 900 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 $100-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 33½%, 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: