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Section 4.5 Value Problems

Objective: Solve value problems by setting up a system of equations.

One application of system of equations are known as value problems. Value problems are ones where each variable has a value attached to it. For example, if our variable is the number of nickles in a person’s pocket, those nickles would have a value of five cents each. We will use a table to help us set up and solve value problems. The basic structure of the table is shown below.

Number Value Total
Item 1
Item 2
Total

The first column in the table is used for the number of things we have. Quite often, this will be our variables. The second column is used for the that value each item has. The third column is used for the total value which we calculate by multiplying the number by the value. For example, if we have 7 dimes, each with a value of 10 cents, the total value is 7Β·10=70 cents. The last row of the table is for totals. We only will use the third row (also marked total) for the totals that are given to use. This means sometimes this row may have some blanks in it. Once the table is filled in we can easily make equations by adding each column, setting it equal to the total at the bottom of the column. This is shown in the following example.

Example 4.5.1.

In a child’s bank are 11 coins that have a value of $1.85. The coins are either quarters or dimes. How many coins each does the child have?

Number Value Total
Quarter q 25
Dime d 10
Total

Using value table, use q for quarters, d for dimes Each quarter’s value is 25 cents, dime’s is 10 cents

Number Value Total
Quarter q 25 25q
Dime d 10 10d
Total

Multiply number by value to get totals

Number Value Total
Quarter q 25 25q
Dime d 10 10d
Total 11 185

We have 11 coins total. This is the number total. We have 1.85 for the final total. Write final total in cents (185). Because 25 and 10 are cents

q+d=11First and last columns are our equations by adding25q+10d=185Solve by either addition or substitutionβˆ’10(q+d)=(11)(βˆ’10)Using addition, multiply first equation byβˆ’10βˆ’10qβˆ’10d=βˆ’110βˆ’10qβˆ’10d=βˆ’110Add together equations25q+10d=185―15q=75Divide both sides by 1515― 15―q=5We have our q, number of quarters is 5(5)+d=11Plug into one of original equationsβˆ’5 βˆ’5―Subtract 5 from both sidesd=6We have our d, number of dimes is 65 quarters and 6 dimes Our Solution

World View Note: American coins are the only coins that do not state the value of the coin. On the back of the dime it says β€œone dime” (not 10 cents). On the back of the quarter it says β€œone quarter” (not 25 cents). On the penny it says β€œone cent” (not 1 cent). The rest of the world (Euros, Yen, Pesos, etc) all write the value as a number so people who don’t speak the language can easily use the coins.

Ticket sales also have a value. Often different types of tickets sell for different prices (values). These problems can be solve in much the same way.

Example 4.5.2.

There were 41 tickets sold for an event. Tickets for children cost $1.50 and tickets for adults cost $2.00. Total receipts for the event were $73.50. How many of each type of ticket were sold?

Number Value Total
Child c 1.5
Adult a 2
Total

Using our value table, c for child, a for adult Child tickets have value 1.50, adult value is 2.00 (we can drop the zeros after the decimal point)

Number Value Total
Child c 1.5 1.5c
Adult a 2 2a
Total

Multiply number by value to get totals

Number Value Total
Child c 1.5 1.5c
Adult a 2 2a
Total 41 73.5

We have 41 tickets sold. This is our number total The final total was 73.50 Write in dollars as 1.5 and 2 are also dollars

c+a=41First and last columns are our equations by adding1.5c+2a=73.5Solve by either addition or substitutionc+a=41We will solve by substitutionβˆ’cβˆ’c―Solve for a by subtracting ca=41βˆ’c1.5c+2(41βˆ’c)=73.5Substitute into untouched equation1.5c+82βˆ’2c=73.5Distribute βˆ’0.5+82=73.5Combine like terms βˆ’82βˆ’82―Subtract 82 from both sides βˆ’0.5c=βˆ’8.5Divide both sides by βˆ’0.5βˆ’0.5β€•βˆ’0.5―c=17We have c, number of child tickets is 17a=41βˆ’(17)Plug into a= equation to find aa=24We have our a, number of adult tickets is 2417 child tickets and 24 adult tickets Our Solution

Some problems will not give us the total number of items we have. Instead they will give a relationship between the items. Here we will have statements such as β€œThere are twice as many dimes as nickles”. While it is clear that we need to multiply one variable by 2, it may not be clear which variable gets multiplied by 2. Generally the equations are backwards from the English sentence. If there are twice as many dimes, than we multiply the other variable (nickels) by two. So the equation would be d=2n. This type of problem is in the next example.

Example 4.5.3.

A man has a collection of stamps made up of 5 cent stamps and 8 cent stamps. There are three times as many 8 cent stamps as 5 cent stamps. The total value of all the stamps is $3.48. How many of each stamp does he have?

Number Value Total
Five f 5 5f
Eight 3f 8 24f
Total 348

Using our value table, f for five cent stamp, and e for eight. Also list value of each stamp under value column.

Number Value Total
Five f 5 5f
Eight e 8 8e
Total 348

Multiply number by value to get total

Number Value Total
Five f 5 5f
Eight e 8 8e
Total 348

The final total was 338 (written in cents) We do not know the total number, this is left blank.

e=3f3 times as many 8 cent stamples as 5 cent stamps5f+8e=348Total column gives second equation5f+8(3f)=348Substitution, substitute first equation in second5f+24f=348Multiply first29f=348Combine like terms29―29―Divide both sides by39f=12We have f. There are 12 five cent stampse=3(12)Plug into first equatione=36We have e, There are 36 eight cent stamps12 five cent, 36 eight cent stamps Our solution

The same process for solving value problems can be applied to solving interest problems. Our table titles will be adjusted slightly as we do so.

Invest Rate Interest
Account 1
Account 2
Total

Our first column is for the amount invested in each account. The second column is the interest rate earned (written as a decimal - move decimal point twice left), and the last column is for the amount of interest earned. Just as before, we multiply the investment amount by the rate to find the final column, the interest earned. This is shown in the following example.

Example 4.5.4.

A woman invests $4000 in two accounts, one at 6% interest, the other at 9% interest for one year. At the end of the year she had earned $270 in interest. How much did she have invested in each account?

Invest Rate Interest
Account 1 x 0.06
Account 2 y 0.09
Total

Use our investment table, x and y for accounts Fill in interest rates as decimals

Invest Rate Interest
Account 1 x 0.06 0.06x
Account 2 y 0.09 0.09y
Total

Multiply across to find interest earned.

Number Value Total
Account 1 x 0.06 0.06x
Account 2 y 0.09 0.09y
Total 4000 270

Total investment is 4000 Total interest was 276

x+y=4000First and last column give our two equations0.06+0.09y=270Solve by either substitution or additionβˆ’0.06(x+y)=(4000)(βˆ’0.06)Use addition, multiply first equation by βˆ’0.06βˆ’0.06xβˆ’0.06y=βˆ’240βˆ’0.06xβˆ’0.06y=βˆ’240Add equations together0.06x+0.09y=270―0.03y=30Divide both sides by 0.030.03―0.03―y=1000We have y$1000 invested at 9%x+1000=4000Plug into original equation βˆ’1000βˆ’1000Subtract 1000 from both sidesx=3000We have x$3000 invested at 6%$1000 at 9% and $3000 at 6%Our Solution

The same process can be used to find an unknown interest rate.

Example 4.5.5.

John invests $5000 in one account and $8000 in an account paying 4% more in interest. He earned $1230 in interest after one year. At what rates did he invest?

Invest Rate Interest
Account 1 5000 x
Account 2 8000 x+0.04
Total

Our investment table. Use x for first rate. The second rate is 4% higher, or x+0.04. Be sure to write this rate as a decimal!

Invest Rate Interest
Account 1 5000 x x
Account 2 8000 x+0.04 8000x+320
Total

Multiply to fill in interest column. Be sure to distribute 8000(x+0.04)

Invest Rate Interest
Account 1 5000 x 5000x
Account 2 8000 x+0.04 8000x+320
Total 1230

Total interest was 1230

5000x+8000x+320=1230Last column gives us our equation13000x+320=1230Combine like terms βˆ’320 βˆ’320―Subtract 320 from both sides13000x=910Divide both sides by 1300013000―  13000―x=0.07We have our x7% interest(0.07)+0.04Second account is 4% higher0.11The account with $8000 is at 11%$5000 at 7% and $8000 at 11%Our Solution

Exercises Exercises – Value Problems

Solve.

1.

A collection of dimes and quarters is worth $15.25. There are 103 coins in all. How many of each is there?

2.

A collection of half dollars and nickels is worth $13.40. There are 34 coins in all. How many of each are there?

3.

The attendance at a school concert was 578. Admission was $2.00 for adults and $1.50 for children. The total receipts were $985.00. How many adults and how many children attended?

4.

A purse contains $3.90 made up of dimes and quarters. If there are 21 coins in all, how many dimes and how many quarters were there?

5.

A boy has $2.25 in nickels and dimes. If there are twice as many dimes as nickels, how many of each kind has he?

6.

$3.75 is made up of quarters and half dollars. If the number of quarters exceeds the number of half dollars by 3, how many coins of each denomination are there?

7.

A collection of 27 coins consisting of nickels and dimes amounts to $2.25. How many coins of each kind are there?

8.

$3.25 in dimes and nickels, were distributed among 45 boys. If each received one coin, how many received dimes and how many received nickels?

9.

There were 429 people at a play. Admission was $1 each for adults and 75 cents each for children. The receipts were $372.50. How many children and how many adults attended?

10.

There were 200 tickets sold for a women’s basketball game. Tickets for students were 50 cents each and for adults 75 cents each. The total amount of money collected was $132.50. How many of each type of ticket was sold?

11.

There were 203 tickets sold for a volleyball game. For activity-card holders, the price was $1.25 each and for noncard holders the price was $2 each. The total amount of money collected was $310. How many of each type of ticket was sold?

12.

At a local ball game the hotdogs sold for $2.50 each and the hamburgers sold for $2.75 each. There were 131 total sandwiches sold for a total value of $342. How many of each sandwich was sold?

13.

At a recent Vikings game $445 in admission tickets was taken in. The cost of a student ticket was $1.50 and the cost of a non-student ticket was $2.50. A total of 232 tickets were sold. How many students and how many non- students attended the game?

14.

A bank contains 27 coins in dimes and quarters. The coins have a total value of $4.95. Find the number of dimes and quarters in the bank.

15.

A coin purse contains 18 coins in nickels and dimes. The coins have a total value of $1.15. Find the number of nickels and dimes in the coin purse.

16.

A business executive bought 40 stamps for $9.60. The purchase included 25c stamps and 20c stamps. How many of each type of stamp were bought?

17.

A postal clerk sold some 15c stamps and some 25c stamps. Altogether, 15 stamps were sold for a total cost of $3.15. How many of each type of stamps were sold?

18.

A drawer contains 15c stamps and 18c stamps. The number of 15c stamps is four less than three times the number of 18c stamps. The total value of all the stamps is $1.29. How many 15c stamps are in the drawer?

19.

The total value of dimes and quarters in a bank is $6.05. There are six more quarters than dimes. Find the number of each type of coin in the bank.

20.

A child’s piggy bank contains 44 coins in quarters and dimes. The coins have a total value of 8.60. Find the number of quarters in the bank.

21.

A coin bank contains nickels and dimes. The number of dimes is 10 less than twice the number of nickels. The total value of all the coins is $2.75. Find the number of each type of coin in the bank.

22.

A total of 26 bills are in a cash box. Some of the bills are one dollar bills, and the rest are five dollar bills. The total amount of cash in the box is $50. Find the number of each type of bill in the cash box.

23.

A bank teller cashed a check for $200 using twenty dollar bills and ten dollar bills. In all, twelve bills were handed to the customer. Find the number of twenty dollar bills and the number of ten dollar bills.

24.

A collection of stamps consists of 22c stamps and 40c stamps. The number of 22c stamps is three more than four times the number of 40c stamps. The total value of the stamps is $8.34. Find the number of 22c stamps in the collection.

25.

A total of $27000 is invested, part of it at 12% and the rest at 13%. The total interest after one year is $3385. How much was invested at each rate?

26.

A total of $50000 is invested, part of it at 5% and the rest at 7.5%. The total interest after one year is $3250. How much was invested at each rate?

27.

A total of $9000 is invested, part of it at 10% and the rest at 12%. The total interest after one year is $1030. How much was invested at each rate?

28.

A total of $18000 is invested, part of it at 6% and the rest at9%. The total interest after one year is $1248. How much was invested at each rate?

29.

An inheritance of $10000 is invested in 2 ways, part at 9.5% and the remainder at 11%. The combined annual interest was $1038.50. How much was invested at each rate?

30.

Kerry earned a total of $900 last year on his investments. If $7000 was invested at a certain rate of return and $9000 was invested in a fund with a rate that was 2% higher, find the two rates of interest.

31.

Jason earned $256 interest last year on his investments. If $1600 was invested at a certain rate of return and $2400 was invested in a fund with a rate that was double the rate of the first fund, find the two rates of interest.

32.

Millicent earned $435 last year in interest. If $3000 was invested at a certain rate of return and $4500 was invested in a fund with a rate that was 2% lower, find the two rates of interest.

33.

A total of $8500 is invested, part of it at 6% and the rest at 3.5%. The total interest after one year is $385. How much was invested at each rate?

34.

A total of $12000 was invested, part of it at 9% and the rest at 7.5%. The total interest after one year is $1005. How much was invested at each rate?

35.

A total of $15000 is invested, part of it at 8% and the rest at 11%. The total interest after one year is $1455. How much was invested at each rate?

36.

A total of $17500 is invested, part of it at 7.25% and the rest at 6.5%. The total interest after one year is $1227.50. How much was invested at each rate?

37.

A total of $6000 is invested, part of it at 4.25% and the rest at 5.75%. The total interest after one year is $300. How much was invested at each rate?

38.

A total of $14000 is invested, part of it at 5.5% and the rest at 9%. The total interest after one year is $910. How much was invested at each rate?

39.

A total of $11000 is invested, part of it at 6.8% and the rest at 8.2%. The total interest after one year is $797. How much was invested at each rate?

40.

An investment portfolio earned $2010 in interest last year. If $3000 was invested at a certain rate of return and $24000 was invested in a fund with a rate that was 4% lower, find the two rates of interest.

41.

Samantha earned $1480 in interest last year on her investments. If $5000 was invested at a certain rate of return and $11000 was invested in a fund with a rate that was two-thirds the rate of the first fund, find the two rates of interest.

42.

A man has $5.10 in nickels, dimes, and quarters. There are twice as many nickels as dimes and 3 more dimes than quarters. How many coins of each kind were there?

43.

30 coins having a value of $3.30 consists of nickels, dimes and quarters. If there are twice as many quarters as dimes, how many coins of each kind were there?

44.

A bag contains nickels, dimes and quarters having a value of $3.75. If there are 40 coins in all and 3 times as many dimes as quarters, how many coins of each kind were there?