Many people believe that mathematics is mainly doing calculations with numbers and formulas – but in reality mathematics has more to do with thinking in a systematic way. Nevertheless, there are some topics in mathematics that do indeed involve a lot of numerical and algebraic computations. Mathematics that we use in everyday financial transactions is like that – there are certain formulas and calculations that you need to master, and there is no way to get around it. It is important to develop a fairly good understanding of these formulas and calculations, so that you can decide for yourself what is in your best financial interest without having to rely on someone else.
Percentages pop up all the time in our financial dealings – in making shopping purchases, computing bank and credit card interest, paying sales and income taxes, calculating discounts and surcharges, and just about everywhere. Anyone without a good basic understanding of percentages in today’s world has not much hope of capably managing his or her finances.
A percentage represents parts per one hundred. For instance, if you take 35% (35 percent) of a pile of rice, that is equivalent to dividing up the rice into 100 equal parts, and then taking 35 of those parts.
If a store clerk charges you 4% sales tax on a $200 purchase, that means the clerk divides the $200 into 100 equal parts of $2 each, and then takes 4 of those parts for a total of $8. Of course, the clerk does not physically divide up the money into 100 equal stacks and take 4 stacks, but instead uses a mathematical formula to accomplish the same result. If A is the amount of a general purchase, and the clerk divides this amount into 100 equal parts, then each part is worth the quotient A/100. When the clerk takes 4 of those parts, the amount of the tax comes to
Thus, to compute 4% of an amount A, you multiply A by .04. The number .04 is the decimal equivalent of 4% – it is obtained by dividing 4 by 100. If we write 4 as 4.0, then dividing 4 by 100 just moves the decimal point two places to the left. To denote that .04 is the decimal equivalent of 4%, we write
4% = .04 .
Naturally, the same reasoning applies to any given percentage – to take 37% of a number you multiply it by .37, and in general to take a percentage p of a number you multiply the number by the decimal equivalent of p, obtained by moving the decimal point two places to the left in p and dropping the percentage sign.
A percentage does not have to be a whole number, nor does it have to be less than 100. If you take 23.5% of a pile of rice, then you are taking 23.5 parts out of 100, and if you take 100% of the rice, you are taking 100 parts out of 100 and hence the whole thing. You can even take 200% of a pile of rice if you find another pile just like it and take both piles – in this case you are taking 200 parts out of 100.
a) If Sandra pays 3.5% sales tax on a grocery purchase of $26, the amount of the tax is
.035 · $26 = $0.91 = 91¢ .
The total amount she must pay is the sum of the purchase price and the tax, or
$26.00 + $0.91 = $26.91 .
You can make this calculation in one step by looking at it this way – she must pay 100% of the purchase price, plus 3.5% of the purchase price for tax, for a total of 103.5% of the purchase price; thus her bill comes to
1.035 · $26 = $26.91 .
b) If Sandra’s credit card company adds a finance charge of 1.32% to her overdue bill of $828.34, then she must pay the additional charge of
.0132 · $828.34 = $10.934088 ≈ $10.93 .
(The wiggly symbol “≈” means “approximately equals”; it indicates that the amount has been rounded off.) The total amount Sandra owes then is
$828.34 + $10.93 = $839.27 .
To do this last calculation in one step, you can add 1.32% to 100% to get 101.32%, and then do the multiplication
1.0132 · $828.34 ≈ $839.27 .
(To add insult to injury, the credit card company might even round up the amount instead of rounding to the nearest cent.)
c) If a purse Sandra likes is priced at $39.95, but is on sale at a 20% discount, then the amount of her discount is
.20 · $39.95 = $7.99 .
The actual price Sandra must pay for the purse is the difference
$39.95 – $7.99 = $31.96 .
The one-step approach is to calculate that Sandra must pay 100% – 20% = 80% of the original price; therefore the discounted price of the purse is
.80 · $39.95 = $31.96 .
To change a decimal back to a percentage, you reverse the previous process, moving the decimal point two places to the right and appending the percentage sign. For instance,
.5 = 50% , 2.64 = 264% , .0775 = 7.75% , .0034 = .34% .
You can change a fraction to a percentage by first converting it to a decimal and then to the equivalent percentage; here are some samples:
|1/2 = .5 = 50%||,||7/5 = 1.4 = 140%||,||3 = 3.00 = 300%|
|5/8 = .625 = 62.5%||,||1/3 ≈ .3333 = 33.33%||,||1/500 = .002 = .2% .|
The calculations illustrate that 1/2 of a quantity is 50% of that quantity, that 7/5 of a quantity is 140% of that quantity, that 3 times a quantity is 300% of the quantity, etc.
You can change a percentage to an equivalent fraction by writing the percentage as parts per one hundred and then reducing the resulting fraction to lowest terms. As illustration, since 25% denotes 25 parts out of 100, we have
25% = 25/100 = 1/4 .
45% = 45/100 = 9/20 ,
37.5% = 37.5/100 = 375/1000 = 3/8 .
(In the last example we multiplied the top and bottom of the fraction by 10 to get rid of the decimal point in the numerator; then we divided top and bottom by the common factor 125 to wind up with 3/8.)
Thus far we have observed that if a number B is a percentage p of another number A, then B, p, and A are related by the equation
B = p · A .
(But before you multiply you have to change p to its decimal equivalent.) Consequently, if any two of the three quantities in this equation are known, you can use the equation to determine the third quantity. The next example illustrates the technique.
Ned got some good deals on his last trip to Foodland.
In each of the questions we will let A denote the original price of the item, p the percentage discount Ned received, and B the monetary amount of his discount.
Part (a) is straightforward; the original price of the potatoes is A = $2.79, the percentage discount is 15%, and thus the monetary discount is
B = p · A = 15% · $2.79 = .15 · $2.79 ≈ $0.42 = 42¢ .
In part (b) we are given that the amount of the discount is B = 78¢ = $0.78 and that the original price of the juice is $1.95. To determine the percentage discount p we plug into the equation B = p · A to get
$0.78 = p · $1.95 .
We divide both sides of the equation by $1.95 to solve for p, and find that
Thus Ned received a discount of 40% on the juice.
Finally, in (c) we are given that the value of the discount is B = 54¢, the percentage discount is p = 30%, and we wish to compute the original price A. Again substituting into the equation B = p · A we find that
54¢ = 30% · A ,
Thus the bananas would cost $1.80 without the discount.
The regular price of a surfboard was $380, but Tim bought it on sale at a 35% discount. How much did he pay for the surfboard?
Because of the discount, instead of paying 100% of the price Tim paid only
100% – 35% = 65% .
Thus the amount he paid for the board was
65% · $380 = .65 · $380 = $247 .
A stylish bowtie, after a discount of 25%, was on sale for only $6.75. What was the original price of the bowtie?
Instead of costing 100% of the original price A, the bowtie cost only 100% – 25% = 75% of that price. Therefore,
$6.75 = 75% · A = .75 · A ,
and the original price was
A = $6.75/.75 = $9.00 .
A new car was advertised at $44,000, but Evelyn was able to negotiate a price of only $40,500. How much of a discount did she receive?
The monetary value of Evelyn’s discount was
$44,000 – $40,500 = $3,500 .
If we let p denote the percentage discount that Evelyn received, then
$3,500 = p · $44,000 ,
and her percentage discount is the quotient
p = $3500/$44000 ≈ .0795 = 7.95% .
Eveyln bargained a discount of almost 8%.
After buying her new car, Evelyn drove it to Macy's at Ala Moana where all women’s clothing was on sale at 20% off. She paid $36 for a groovy shirt, $44 for some eye-catching slacks, and $56 for a snazzy pair of boots. How much money did she save by paying the discount prices?
The grand total for all of Evelyn’s purchases was the sum
$36 + $44 + $56 = $136 .
The percentage of the original price of the three items that she paid was
100% – 20% = 80% .
If we let A represent the original price of all the items, then
$136 = 80% · A = .80 · A
A = $136/.80 = $170 .
The amount that Evelyn saved was the difference between the original price and the sale price – namely,
$170 – $136 = $34 .
Glenn saved $135 on a riding lawn mower by buying it on sale at 15% off the regular price. What was the regular price, and how much did Glenn pay for the mower?
Glenn’s savings of $135 represents 15% of the regular price of the mower. Denoting the regular price as A, we deduce that
$135 = 15% · A = .15 · A ,
A = $135/.15 = $900 .
Glenn paid the difference between the regular price and the discount,
$900 – $135 = $765 .
Sometimes if you are lucky you might be awarded a chain discount – that is, a discount added to a discount. Suppose for example that you work for Uyeda shoe store, and the store is giving a special weekend discount of 20% on all its shoes. Suppose also that as an employee of the store you are entitled to an additional 10% discount on all merchandise. How much will you have to pay for a funky pair of loafers whose regular price is $60?
You might first guess that your total discount on the shoes will amount to the sum 20% + 10% = 30%. However, the second discount of 10% does not apply to the original price, but only to the discounted price after the first 20% discount – consequently your total discount will be somewhat less than 30% of the original price.
Let us carry out the calculations. After the first 20% discount, the price of the shoes will be 80% of the original price, or
80% · $60 = .80 · $60 = $48 .
Next, after the 10% discount, the price that you pay for the shoes will be 90% of the first discounted price, or
90% · $48 = .90 · $48 = $43.20 .
Your total savings on the shoes is $60.00 – $43.20 = $16.80. The percent of the original price that you saved is the quotient
$16.80/$60.00 = .28 = 28% .
Our conclusion is that a chain of two discounts of 20% and 10% is equivalent to a single discount of 28%.
We could reach the same conclusion even without knowing the price of the item being purchased. Let us suppose that the original price of the item is P. After the first 20% discount the price reduces to 80% of P. Then after the second discount of 10% you must pay 90% of the first discounted price, or 90% of 80% of P. This the cost of the item after the two discounts is
90% of 80% of P = .90 · .80 · P = .72 · P = 72% of P .
Since you must pay 72% of the original price of the item, your total percentage discount is 100% – 72% = 28%.
|a. 2/5||b. 5/6||c. 5/2||d. 3/8||e. 1/4000|
|a. 90%||b. 85%||c. 225%||d. 4%||e. 87.5%|