Блог Анкара

In This Chapter

^ Working with denominations and number of coins

^ Figuring out the total amount of money in coins or bills

^ Working with money from around the world

Oney is a common denominator. We deal with money just about every day — from the spending end, the earning end, or both. Counting coins is one of the earliest number exercises for a small child. And collecting coins is a passion of many historians and numismatists.

This chapter deals with computing totals of money and figuring out the number of coins from the totals. You even get a short primer on monetary units from several different countries and see how the basic properties are the same for just about any monetary system.

Determining the Total Count

Counting up the coins in your pocket or from your piggy bank involves more than just knowing that you have 80 coins. The number of coins doesn’t tell you how much money you have. You usually have different denominations, so you need to count each type of coin differently. Each type of coin gets multiplied by its monetary value.

Equating different money amounts

Each country has its own monetary system with its set of coins and other currency. Often, the money is imprinted with pictures of historic figures and places. In the United States, for example, each state is honored (or soon will

Be) with its own quarter. What is similar to the coinage of different countries is that different coins take on different values. In the United States, it takes more nickels to make a dollar than it does quarters, but it sure feels like you have more money when you have a dollar’s worth of nickels rather than a dollar’s worth of quarters.

The Problem: You want to change your $5 bill into nickels or quarters. How many nickels and how many quarters are needed to equal $5?

£,?LAiV A nickel is worth 5<t, and a quarter is worth 25<t. Five dollars is equal to 500<t. To determine the number of nickels in $5, divide 500<t by 5<t: 500 5 = 100 nickels. The number of quarters is determined by dividing 500 cents by 25 cents: 500 ■ 25 = 20 quarters.

The Problem: A roll of quarters from the bank equals $10, a roll of dimes equals $5, a roll of nickels equals $2, and a roll of pennies equals 50<t. How many rolls of dimes is worth the same as 20 rolls of nickels?

«*,VLA/V Sort through the information, first. You only need to know about the dimes and nickels. Determine the total of 20 rolls of nickels. Then figure out how many rolls of dimes you can get for that amount of money. Twenty rolls of nickels is 20 x $2 = $40. Divide 40 by 5: $40 $5 = 8 rolls of dimes.

The Problem: A roll of dimes is worth $5, a roll of nickels is worth $2, and a roll of pennies is worth 50<t. Which is worth more: 10 rolls of nickels, 4 rolls of dimes, or 50 rolls of pennies?

Multiply the number of rolls of nickels by $2, the number of rolls of dimes by $5, and the number of rolls of pennies by half a dollar, or 50<t. (An alternative would be to change the $2 and $5 to cents; in any case, the units all have to be the same to make a comparison.) Doing the calculations: 10 rolls of nickels x $2 = $20; 4 rolls of dimes x $5 = $20; 50 rolls of pennies x $0.50 = $25. The pennies are worth more than the nickels or dimes by $5.

Adding it all up

To find the total amount of money you have in coins or bills, you multiply the number of each by their monetary value and add up all the products for the total.

The Problem: You open up your piggy bank and find 177 pennies, 123 nickels, 59 dimes, 33 quarters, and 5 half-dollars. What is the total amount of money in your piggy bank?

Multiply the number of pennies by 1, nickels by 5, dimes by 10, quarters by 25, and half-dollars by 50. This gives you the number of cents. It’s easier to multiply by whole numbers, at first, and then change to dollars by moving the decimal point two places to the left. Doing the computation:

177 x 1$ = 177$ 123 x 5$ = 615$ 59 x 10$ = 590$ 33 x 25$ = 825$ 5 x 50$ = 250$

The sum of the products is: 177 + 615 + 590 + 825 + 250 = 2,457$. Moving the decimal point two places (or dividing by 100), you get that the total in the piggy bank is $24.57.

The Problem: You are in charge of the concession stand at Friday’s football game. You need to bring enough coins and bills to make change for the purchases of the customers. You’re given a check for $200 to pay for the change and decide to get 50 $1 bills, 20 $5 bills, and the rest of the change in quarters and nickels. Quarters come in rolls worth $10, and nickels come in rolls worth $2. How many rolls of quarters and nickels can you get?

You first determine how much money in coins you’ll be getting after taking care of the $1 bills and $5 bills. Add up the total for the $1 bills and $5 bills. Then subtract the total of the bills from $200. Many different combinations of rolls of quarters and nickels add up to the amount you’ll need in coins. Make a chart, putting the number of rolls of quarters in one column, its worth in the next column, the value remaining in the third column (after spending this much on the quarters), and then the number of rolls of nickels you can buy in the last column.

Ancient coin

An archaeologist claims that he found an all people who use eBay are much too smart for ancient coin dated 46 B. C. He put it on eBay and him. Why didn’t anyone bid on the coin? hoped to sell it for a small fortune. Fortunately,

■81UIJ 84J JE BuiSn SEM 84 JEpU8|E0 J8A8JE4M Uo pOSEq JEeA

E uo jnd 8AE4 P|noM pus -o-a sem ji JE4J Umou>| 8ae4 j, up|noM moo 341 PojE – ojo J3A304M -o-a 9fr ujoj)A||eoj sem moo eqj(| jeA pemuuejep uooq j, uPE4 ■a-v pus -o-a ueeMjeq sjeoA eqj ui >|Eejq eqj -o-a St J=eA eqj u, :i9MsuV

Determining the amount in bills: 50 x $1 = $50 and 20 x $5 = $100. The total of $150 leaves $50 in coins ($200 – $150 = $50). In Table 9-1, I filled in the number of rolls of quarters first. Then I computed the value of the quarters, leaving the amount for nickels. Finally, I computed the number of rolls of nickels.

You’ll need no quarters with 25 rolls of nickels, 1 roll of quarters and 20 rolls of nickels, 2 rolls of quarters with 15 rolls of nickels, 3 rolls of quarters and 10 rolls of nickels, 4 rolls of quarters and 5 rolls of nickels, or 5 rolls of quarters with no nickels.

Working Out the Denominations of Coins

If you have enough different kinds of coins, you can create any number of money amounts. Some money amounts are possible with several different combinations of coins. You want to be creative. The standard U. S. coins are used in the problems in this section: penny (1*), nickel (5*), dime (10*), quarter (25*), half-dollar (50*), and dollar (100* or $1).

Having the total and figuring out the coins

Some people like to have lots of coins in their pockets. It gives them a feeling of wealth. Others don’t like to have their pockets bulge too much, so they prefer not to have a lot of extra coins around. There are problems in this section for both types of people — depending on which type you are.

The Problem: Stefanie wants to give each of her children 42*. How can she do this with the least number of coins for each child?

Two strange coins

A Certain Country has just two kinds of coins. One coin is caWed a sevna and is worth 7 units. The other coin is a levna, and it’s worth 11 units. Because of these strange units, some prices can’t be paid for exactiy using these coins.

Residents can pay for items costing 7 units or 14 units or 18 units, and so on. What is the highest priced item that can’t be paid for with any combination of these coins?

■seunei PuE Seunes Qj|Mjoj pied eq ueo 6S ueqj jeBBiq jeqiunu Auv ’6S Jaqw™ eqj si su pue st jo uoij – Emqiuoo b jo u jo L Jo 6|d|j|niu e j. usi JEqj uieji peoud jseqbiq eqi ‘j. usi jl jnq ‘jo, pied eq j. ueo JEqj eoud q6|q Ajjejd b s, j| jBqj >|u! qj Ablu noAUwsuv

Make a chart of the different denominations of coins and the number of each type of coin that is needed to add up to 42*. The table won’t have to contain 50-cent pieces or dollars, because they’re both worth more than the total needed. See the chart I created in Table 9-2.

I’ll stop here with the table. The numbers aren’t getting any better. It looks like one quarter, one dime, one nickel, and two pennies are the fewest number of coins needed to add up to 42*.

The Problem: How many different ways can you make change for a quarter?

Haul out another chart. As you fill in the different amounts, do this in an organized fashion, using dimes as many ways as you can, nickels as many ways as you can, and so on, just to make it less likely that you’ll miss something. Table 9-3 has just the totals for each coin, not the number of coins.

As you see, you can make change for a quarter in 12 different ways, using standard U. S. coins.

The Problem: How can you have a total of $1.40 using exactly 24 coins?

You can do this problem with a chart — a Big Chart. But another, not-too-glamorous way, is to try to work it out through trial and error and reasoning. For example, you could try using five quarters (the greatest number of quarters possible), leaving 140* – 125* = 15 cents. Even if you used all pennies for the rest of the coins, that’s only a total of 5 + 15 = 20 coins, which isn’t enough, if you’re aiming for 24 coins.

Next, you could try using four quarters, leaving 140 – 100 = 40*. You now need to use 20 more coins to add up to 40*. One dime and 30 pennies is too many coins. Two dimes and 20 pennies is still too many coins. Three dimes and ten pennies is too few coins. Are you ready to go to a chart yet? No, I’m not giving up that easily.

Think about the pennies. If you’re going to use pennies, they have to be a multiple of 5 for the total to come out to an even 140*. You need to use 5, 10, 15, or 20 pennies so that the other coins will add up with them and come out right. Start with 20 pennies (the most you can use without exceeding 24

Coins) and work downward. Using 20 pennies, you now need 4 more coins to add up to $1.20. You can do this in two different ways: a dollar coin plus a dime and two nickels, or two 50-cent pieces plus two dimes. Whew! Two answers! And there are more. For example, 2 quarters plus 8 dimes plus 9 nickels plus 5 pennies are 24 coins that add up to $1.40. Can you find any more? If so, e-mail me at Sterling@bradley. edu, and I’ll post them on my Web site (Http://hilltop. bradley. edu/~sterling/facinfo. html)!

Going with choices of coins and bills

The coins used in the United States have several things in common with the coins used in other countries. The biggest commonality is having 100* in $1. Most countries have monetary systems with 100 of some coin being equal to the main monetary unit.

In the metric system, each unit is ten times or one-tenth of another unit. Not so in our money system. Even with five nickels in a quarter and ten dimes in a dollar, there is still no way to divide a quarter into an equal number of dimes. Our paper currency is a bit more forgiving, but there are still challenges with the dividing and multiplying.

The Problem: How many nickels, dimes, quarters, and half-dollars are equal in value to $5?

Divide 500* (5 x 100*) by the cents value of each of the coins. For nickels, 500* 5* = 100 nickels. With dimes, 500* 10* = 50 dimes. Quarters give you 500* 25* = 20. And half-dollars yield 500* 50* = 10 half-dollars.

The Problem: How many quarters are equivalent to $1, $2, $5, $10, $20, $50, AiASfyy and $100?

Sliding the pennies

Look at the triangle of pennies shown in the figure. Can you reverse the triangle so that it points downward instead of upward — and do it by moving only three pennies?

■6 pus 8 somuod mo|oq ‘unnop t Auuod oaoui uou.I ■E Auuad oj jxsu dn 0l Auuod pus Z Auuod oj jxou dn L Auuod oaowNa/moiy

^VLA* If you divide $1 by 25*, that’s 100 25 = 4 quarters. Two dollars divided by 25* is 200 25 = 8 quarters. You can do the divisions for the rest of the bills, but an easier way than dividing by 25 is multiplying by 4. Because $1 is equal to 4 quarters, $20 is equal to 20 x 4 = 80 quarters, $50 is equal to 50 x 4 = 20 quarters, and so on.

Figuring Coins from around the World

Most of the countries in the world have coin systems in which one of the coins is worth 100 of another. Other coins are then multiples of that smaller coin — usually 10 times, 20 times, or 50 times. The exceptions are few. The coin names are usually pretty interesting, though, and probably have some historical significance.

Making change in another country

In India, the monetary unit is the rupee. You find banknotes of 5, 10, 20, 50, and 100 rupees. There are also 1-rupee, 2-rupee, and 5-rupee coins. The smaller coins in India are paise. One rupee is equal to 100 paise. The coins that are multiples of the paise are 10-paise coins, 20-paise coins, 25-paise coins, and 50-paise coins.

The Problem: Is there a single coin in India that is equal to: three rupees, one 50-paise coin, three 20-paise coins, three 25-paise coins, and 15 paise? (In other words, if you have all that in your pocket, can you exchange it for a single coin?)

^VLA/rV Determine the total amount of money all these coins are worth. Then look at the equivalence between rupees and paise and see if a coin in the list is equal to the sum. Just concentrate on the paise, at first. When you get a total, you can see if you’ll need to change the rupees to paise or paise to rupees. Multiplying each of the paise coins by how many of each you have: (1 x 50) + (3 x 20) + (3 x 25) + (15 x 1) = 50 + 60 + 75 + 15 = 200 paise. Because 100 paise equals 1 rupee, the 200 paise are worth 2 rupees. Add these two rupees to the three rupees you already have, and you can exchange the five 1-rupee coins for a 5-rupee coin.

The Problem: Some interesting Indian coins are no longer used. In the late 1940s, you got 16 annas for 1 rupee, 4 pice for 1 anna, and 3 pies for 1 pice. How many pies could you get for 12 rupees?

Pushing a coin through paper

Can you push a quarter through a dime-sized in the paper. Now push a quarter through the hole in a piece of paper? Take a dime and trace hole without tearing the paper. How can you do around it on a piece of paper. Then carefully cut this? out along the circle you’ve drawn, leaving a hole

■jno peuejjE|j ueeq seq jeqj 8|oq eqj q6nojqj di|s ||im jejjenb eqi ■jeqjo qoee Pjemoj ‘unnop (jeded eqj pue] weqj pueq pue p|oj eqj jo spue eqj e>|Ej ueqi -9|oM eqj qBnojqj seoB p|oj eqj jeqj os jeded eqj pioj.-ja/wsu^

Multiply the number of rupees by 16 to get the number of annas. Multiply that product by 4 to get the number of pies. Then multiply that product by 3 to get the number of pies.

16 annas 4 pice 3 pies

12 rupees X ^-x ^—^-x Tj——.—

1 rupee 1 anna 1 pice

Iri.,16 annas 4 pice 3 pies = 12 rupees X.^x.^ x, , 1 rupee 1 anna 1 pice

= 12 X 16 X 4 X 3 pies = 2,304 pies

The Problem: In China, the monetary unit is yuan. One yuan is equal to 10 jiao, and 1 jiao is equal to 10 fen. The multiples and powers of ten are at work here, making the computation much easier. What is the fewest number of coins (or paper bills) you need if you have 6,348 fen?

^VLA/rV First determine how many fen in a yuan. Then divide 6,348 by that number to get as many yuan as possible. Next, divide the remainder by 10 to convert them to jiao. The rest will have to be in fen, and there should be fewer than 10 fen. If there are 10 fen in 1 jiao and 10 jiao in 1 yuan, then there are 10 X 10 fen in 1 yuan. Divide 6,348 by 100 to get 63 yuan with a remainder of 48. Divide 48 by 10 to get 4 jiao with 8 left over. 63 yuan + 4 jiao + 8 fen is equal to 75 coins or bills.

Converting other currency to U. S. dollars

A common challenge for people traveling in foreign countries is converting their money to the currency of that country and back again. If you’re unfamiliar with a particular monetary unit, you need to get acquainted with the relative

Value (relative to your money) so that you know what the worth is of what you’re buying. The exchange rates change daily, so the problems in this section use approximate values to give you an idea of the different conversions.

The Problem: If one U. S. dollar is equal to 8.28 Chinese yuan, then what is the approximate cost, in dollars, of a silk robe that’s selling for 3,000 yuan?

The biggest challenge to doing these problems is in deciding whether to multiply or divide — and by what. A good approach to doing money conversions is to use proportions. Chapter 7 goes into more detail on the properties of proportions. In the case of dollars and yuans, write a fraction with $1 divided by 8.28 yuan. Then set that fraction equal to X Dollars divided by 3,000 yuan. Note that the dollars are opposite dollars, in the numerator, and the yuans are opposite yuans in the denominator.

1 dollar

X Dollars

8.28 yuan 3,000 yuan Now cross-multiply and solve for X.

3000 = 8.28X 3000 828x

8.28

362.3188

8-28

X

The silk robe costs about $362.

VLAiV

The Problem: In Vietnam, the monetary unit is the dong. One dong is equal to 10 hao, and 1 hao is equal to 10 xu. About 15,300 dong are equivalent to $1. You have been traveling in Vietnam and have spent the day with a wonderful tour guide. You want to tip him accordingly and figure that an extra $40 (American) will do that to your satisfaction. How many dong will you add to the bill for his tip?

Set up a proportion with dong and dollars. Solve for the needed number of dong.

1 dollar 40 dollars

——- =—

15,300 dong X Dong X = 15,300 X 40 = 612,000

You will be tipping your guide 612,000 dong.

Chapter 10