Algebra · Chapter 19

System of Equations

Name the unknowns, turn every fact into an equation, and combine them: systems, mixtures and work-rate problems the contest way.

What's this chapter about?

19.0 Setting Up Systems
19.1 Work and Rate
19.2 Practice Problems

You'll need

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19.00%

Setting Up Systems

In this section you will be able to name the unknowns in a word problem, turn each fact into an equation, and solve the system by elimination or substitution.

Team A won 2/3 of its games, team B won 5/8. B won 7 more and lost 7 more. Two unknowns, two facts: that is a system.

Every problem in this chapter follows the same three moves. Name the unknowns with letters. Translate each sentence of the problem into one equation. Combine the equations until one unknown is left.

Two equations in two unknowns are two lines; the answer is where they cross. To combine them, eliminate (add or subtract whole equations so a letter cancels) or substitute (solve one equation for a letter and plug it into the other).

Look:Often you never need every unknown. Ask what the question wants, and aim straight for it.

Exploreplay first, formulas later

Try this: Name the unknowns first: which two numbers does Example 19.1 hide? Here a counts team A's games and b counts team B's.
Try this: Each line is one fact: wins or losses. Guess roughly where they cross. Then solve Example 19.1 below and turn on "Show the crossing" to check.
Try this: Drag a coefficient until the lines are parallel. How many solutions now? What if the two lines lie on top of each other?

Watch itpredict, then reveal

Example 19.1

AMC 10

Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\frac{2}{3}$ of its games and team $B$ has won $\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A$ . How many games has team $A$ played?

1
Name the unknowns first. Let team $A$ play $a$ games and team $B$ play $b$ games. How many games has $A$ won, and how many has it lost?

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Try it

Problem 19.2.1

AMC 10

In an after-school program for juniors and seniors there is a debate team with an equal number of students from each class on the team. Among the $28$ students on the program, $25\%$ of the juniors and $10\%$ of the seniors are on the debate team. How many juniors are in the program?

Problem 19.2.2

AMC 10

Joe has a collection of $23$ coins, consisting of $5$ -cent coins, $10$ -cent coins, and $25$ -cent coins. He has $3$ more $10$ -cent coins than $5$ -cent coins, and the total value of his collection is $320$ cents. How many more $25$ -cent coins does Joe have than $5$ -cent coins?

Problem 19.2.4

AMC 10

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

Contest cornerreal contest problems

Problem 19.2.3

AMC 10

The sum of two natural numbers is $17{,}402$ . One of the two numbers is divisible by $10$ . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

Problem 19.2.5

AMC 10/12

At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?

Problem 19.2.6

AMC 10/12

When a student multiplied the number $66$ by the repeating decimal

1.\underline{a}\,\underline{b}\,\underline{a}\,\underline{b}\cdots = 1.\overline{\underline{a}\,\underline{b}},

where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.\underline{a}\,\underline{b}$ . Later he found that his answer is $0.5$ less than the correct answer. What is the $2$ -digit integer $\underline{a}\,\underline{b}$ ?

Problem 19.2.8(additional)

AMC 10

Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n$ ?

Problem 19.2.11(additional)

AMC 10

In the United States, coins have the following thicknesses: penny, $1.55$ mm; nickel, $1.95$ mm; dime, $1.35$ mm; quarter, $1.75$ mm. If a stack of these coins is exactly $14$ mm high, how many coins are in the stack?

Problem 19.2.14(additional)

AMC 12

The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$ . Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?

Checkpoint for section 19.0

Get 2 of 3 right to master this section. One try each, no hints.

Joe's three coin facts from Problem 19.2.2 are drawn as three scales ( $n$ nickels, $d$ dimes, $q$ quarters). Find values that balance all three scales at once. How many dimes does Joe have?

Numbers $x$ , $y$ , $z$ satisfy $2x + y + z = 36$ , $x + 2y + z = 40$ and $x + y + 2z = 32$ . What is $x + y + z$ ?

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?

19.10%

Work and Rate

In this section you will be able to use Work = Rate × Time, add rates when people work together, and split a job into stages when the team changes.

Abe paints alone, then Bea joins, then Coe. When is the room done? Rates add, times don’t.

A rate is how much of the job gets done per unit of time. If you finish a job in $a$ hours, your rate is $\frac{1}{a}$ job per hour. When people work together, their rates add.

When the team changes halfway, split the job into stages. In each stage the rate is constant, so Work = Rate × Time works stage by stage.

Look:For a factory problem, measure the work itself in worker-hours: 60 workers for 2 hours is 120 worker-hours.

Exploreplay first, formulas later

Try this: Turn both faucets on. Predict the fill time with ab/(a + b), then press "When is it full?" to compare.
Try this: Each faucet's rate is 1/a or 1/b of a tub per hour. Show the formula: why do we add rates, not times?
Try this: Change faucet b to 12 hours. Does the pair get faster or slower than faucet a alone? Why must it always be faster?

Name it

Theorem 19.1.1 · Work, Rate and Time

\text{Work} = \text{Rate} \times \text{Time}

Equivalently,

\text{Rate} = \frac{\text{Work}}{\text{Time}}, \qquad \text{Time} = \frac{\text{Work}}{\text{Rate}}.

Theorem 19.1.2 · Working Together

If one person can do something in $a$ amount of time, and someone else can do the same thing in $b$ amount of time, working together they can do it in

\frac{ab}{a+b}

time.

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Example 19.2

AIME

Abe can paint the room in $15$ hours, Bea can paint $50$ percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.

1
Work in rooms per hour. Abe paints $\frac{1}{15}$ room per hour. What are Bea's and Coe's rates?

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Example 19.3

AIME

The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can produce $240$ widgets and $300$ whoosits. In three hours, $50$ workers can produce $150$ widgets and $m$ whoosits. Find $m$ .

1
Measure work in worker-hours. Let one widget take $w$ worker-hours and one whoosit $h$ . How many worker-hours does the first fact give?

Predict first:
Your prediction

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Contest cornerreal contest problems

Problem 19.2.15(additional)

AMC 10/12

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 PM. How long, in minutes, was each day's lunch break?

Problem 19.2.18(additional)

AIME

In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?

Problem 19.2.19(additional)

AIME

A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.

Checkpoint for section 19.1

Get 2 of 3 right to master this section. One try each, no hints.

19.2

Practice Problems

In this section you will be able to mix systems, mixtures, invariants and work-rate on harder contest problems.

Time to mix it all up: averages, marbles, coin machines, acid jars and a system that hides a triangle. Name the unknowns first, every time.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 19.2.7

AMC 10B/12B

This problem also appears in the book as 15.1.3.

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then returned to the set, the average value of the integers rises to $40$ . The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$ . What is the average value of all the integers in the set $S$ ?

Problem 19.2.9(additional)

AMC 10

Problem 19.2.10(additional)

AMC 10

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$ , and the ratio of blue to green marbles in Jar $2$ is $8:1$ . There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$ ?

Problem 19.2.12(additional)

AMC 10

Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

Problem 19.2.13(additional)

AMC 10

One morning each member of Angela's family drank an $8$ -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

Problem 19.2.16(additional)

AMC 12

A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?

Problem 19.2.17(additional)

AIME

Jar A contains four liters of a solution that is $45\%$ acid. Jar B contains five liters of a solution that is $48\%$ acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are $50\%$ acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$ .

Problem 19.2.20(additional)

AIME

Given that $x$ , $y$ , and $z$ are real numbers that satisfy:

x = \sqrt{y^2 - \frac{1}{16}} + \sqrt{z^2 - \frac{1}{16}},

y = \sqrt{z^2 - \frac{1}{25}} + \sqrt{x^2 - \frac{1}{25}},

z = \sqrt{x^2 - \frac{1}{36}} + \sqrt{y^2 - \frac{1}{36}},

and that $x + y + z = \frac{m}{\sqrt{n}}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m + n$ .