1.9: Rational Equations

1.9 Rational Equations
We can also use the skills we have covered in previous sections to solve equations involving rational expressions. There are three main methods of solution that will be explored in this section - multiplying on both sides to clear a denominator, cross-multiplying, and making common denominators. Each of these techniques is actually the same process, but approached from a slightly different perspective.
Clearing a denominator

Often, if the denominator is a single variable, it can be easy and striaghtforward to multiply on both sides by the denominator to cancel it out.

Example
Solve for \(x\)
\[
x+\frac{4}{x}=7
\]
If we multiply on both sides by \(x\), it will clear the variable from the denominator:
\[
\begin{array}{c}
x+\frac{4}{x}=7 \\
x *\left(x+\frac{4}{x}\right)=(7) * x \quad \text { Multiply on both sides by } x
\end{array}
\]
\[
x * x+x * \frac{4}{x}=7 x \quad \text { Distribute the } x
\]
\[
\begin{array}{c}
x^{2}+4=7 x \\
x^{2}-7 x+4=0
\end{array} \quad \text { Standard form }
\]
\[
x \approx 0.628,6.372
\]

If a problem is stated simply as the equality of two fractions, cross-multiplying can be a useful method of solution.

Example
Solve for \(x\)
\[
\begin{aligned}
\frac{x}{2 x+3}=\frac{7}{x-4} \\
\frac{x}{2 x+3} &=\frac{7}{x-4} \\
x(x-4) &=7(2 x+3) \\
x^{2}-4 x &=14 x+21 \\
x^{2}-18 x-21 &=0 \\
x & \approx 19.100,-1.100
\end{aligned}
\]
Cross-multiplying is really just a short-cut method of clearing out the denominators by multiplying on both sides by both denominators:
\[
\begin{aligned}
\frac{x}{2 x+3} &=\frac{7}{x-4} \\
(x-4)(2 x+3) * & \frac{x}{2 x+3}=\frac{7}{x-4} *(x-4)(2 x+3) \\
(x-4)\cancel{(2 x+3)} * & \frac{x}{\cancel{2 x+3}}=\frac{7}{\cancel{x-4}} *\cancel{(x-4)}(2 x+3) \\
x(x-4) &=7(2 x+3)
\end{aligned}
\]
Then the equation is ready to be solved as shown above - but by just cross-mulitiplying, we skip directly to the solution portion of the problem.

Sometimes, it is helpful to create a common denominator in order to set up a situation where cross-multiplying can be used.

Example
Solve for \(x\)
\[
\begin{aligned}
\frac{1}{x+6}+\frac{4}{x-2}=\frac{3}{x+1} & \\
\frac{1}{x+6}+\frac{4}{x-2}=\frac{3}{x+1} \\
\frac{1}{x+6} * \frac{x-2}{x-2}+\frac{4}{x-2} * \frac{x+6}{x+6}=\frac{3}{x+1} \\
\frac{1(x-2)+4(x+6)}{(x+6)(x-2)}=& \frac{3}{x+1} \\
\frac{x-2+4 x+24}{(x+6)(x-2)}=& \frac{3}{x+1} \\
\frac{5 x+22}{(x+6)(x-2)}=& \frac{3}{x+1} \\
(5 x+22)(x+1)=3(x+6)(x-2)=3\left(x^{2}+4 x-12\right) & \\
5 x^{2}+27 x+22=3 x^{2}+12 x-36 & \\
2 x^{2}+15 x+58 &=0 \\
x \approx-3.75 \pm 3.865 i &
\end{aligned}
\]

Example
Solve for \(x\)
\[
\begin{array}{c}
\frac{2}{x-2}+\frac{x}{2 x-1}=4 \\
\frac{2}{x-2}+\frac{x}{2 x-1}=4 \\
\frac{2}{x-2} * \frac{2 x-1}{2 x-1}+\frac{x}{2 x-1} * \frac{x-2}{x-2}=4 \\
\frac{2(2 x-1)+x(x-2)}{(x-2)(2 x-1)}=4 \\
\frac{4 x-2+x^{2}-2 x}{(x-2)(2 x-1)}=4 \\
\frac{x^{2}+2 x-2}{(x-2)(2 x-1)}=\frac{4}{1} \\
1\left(x^{2}+2 x-2\right)=4(x-2)(2 x-1)=4\left(2 x^{2}-5 x+2\right) \\
x^{2}+2 x-2=8 x^{2}-20 x+8 \\
\qquad \begin{aligned}
\frac{2}{x} &=7 x^{2}-22 x+10 \\
x &=2.592,0.551
\end{aligned}
\end{array}
\]

In some situations, we can create a single common denominator for every fraction in the problem and then clear them all at once.

Example
Solve for \(x\)
\[
\begin{aligned}
\frac{2}{x+3}-\frac{4}{3 x-1}=\frac{x}{3 x^{2}+8 x-3} & \\
\frac{2}{x+3}-\frac{4}{3 x-1} &=\frac{x}{3 x^{2}+8 x-3} \\
\frac{2}{x+3} * \frac{3 x-1}{3 x-1}-\frac{4}{3 x-1} * \frac{x+3}{x+3} &=\frac{x}{3 x^{2}+8 x-3} \\
\frac{2(3 x-1)-4(x+3)}{(x+3)(3 x-1)} &=\frac{x}{(x+3)(3 x-1)} \\
2(3 x-1)-4(x+3) &=x
\end{aligned}
\]
The missing step above is the clearing of both denominators:
\[
(x+3)(3 x-1) * \frac{2(3 x-1)-4(x+3)}{(x+3)(3 x-1)}=\frac{x}{(x+3)(3 x-1)} *(x+3)(3 x-1)
\]
\[
\begin{aligned}
\cancel{(x+3)}\cancel{(3 x-1)} * \frac{2(3 x-1)-4(x+3)}{\cancel{(x+3)}\cancel{(3 x-1)}} &=\frac{x}{\cancel{(x+3)}\cancel{(3 x-1)}} *\cancel{(x+3)}\cancel{(3 x-1)} \\
& 2(3 x-1)-4(x+3)=x
\end{aligned}
\]

As is true in the process of cross-multiplying, it isn't necessary to actually cancel out the denominators in completing the problem.
\[
\begin{aligned}
\frac{2}{x+3}-\frac{4}{3 x-1} &=\frac{x}{3 x^{2}+8 x-3} \\
\frac{2}{x+3} * \frac{3 x-1}{3 x-1}-\frac{4}{3 x-1} * \frac{x+3}{x+3} &=\frac{x}{3 x^{2}+8 x-3} \\
\frac{2(3 x-1)-4(x+3)}{(x+3)(3 x-1)} &=\frac{x}{(x+3)(3 x-1)} \\
2(3 x-1)-4(x+3) &=x \\
6 x-2-4 x-12 &=x \\
2 x-14 &=x \\
x &=14
\end{aligned}
\]

Exercises 1.9
1) \(\quad x+\frac{5}{x}=-6\)
2) \(\quad x+\frac{6}{x}=-7\)
3) \(\quad y-\frac{5}{y}=2\)
4) \(\quad \frac{7}{a}+1=a\)
5) \(\quad \frac{9}{2 y+4}=\frac{3}{y}\)
6) \(\quad \frac{4}{3 n+7}=\frac{1}{2}\)
7) \(\quad \frac{x}{x+3}=\frac{8}{x+6}\)
8) \(\quad \frac{y-2}{2}=\frac{5}{y-5}\)
9) \(\quad \frac{2}{n}=\frac{n}{5 n+12}\)
10) \(\quad \frac{x}{4-x}=\frac{2}{x}\)
11) \(\quad \frac{5 x}{14 x+3}=\frac{1}{x}\)
12) \(\quad \frac{a}{8 a+3}=\frac{1}{3 a}\)
13) \(\quad \frac{9}{x-1}-\frac{2}{x+4}=\frac{1}{x+2}\)
14) \(\quad \frac{1}{x-2}+\frac{4}{x+5}=\frac{1}{x-3}\)
15) \(\quad \frac{5}{3 x+2}+\frac{1}{x-1}=\frac{3}{x+2}\)
16) \(\quad \frac{1}{y-2}-\frac{4}{2 y+5}=\frac{6}{y-1}\)
17) \(\quad \frac{5}{x+1}+\frac{1}{x+2}=3\)
18) \(\quad \frac{1}{2 x-1}-\frac{2}{x+7}=1\)
19) \(\quad \frac{6}{y-4}-\frac{1}{y+2}=3\)
20) \(\quad \frac{10}{a+1}+\frac{3}{a-2}=2\)
21) \(\quad \frac{3 a}{a^{2}-2 a-15}-\frac{a}{a+3}=\frac{2 a}{a-5}\)
22) \(\quad \frac{u^{2}+2}{u^{2}+u-2}-\frac{3 u}{u+2}=\frac{-2 u-1}{u-1}\)
23) \(\quad \frac{4}{2 x-1}+\frac{2}{x+3}=\frac{5}{2 x^{2}+5 x-3}\)
24) \(\quad \frac{5}{x+5}-\frac{2}{x^{2}+2 x-15}=\frac{2}{x-3}\)
25) \(\quad \frac{5}{y-2}-\frac{3}{2 y-1}=\frac{4}{2 y^{2}-5 y+2}\)
26) \(\quad \frac{x+2}{x-1}+\frac{x+4}{x}=\frac{2 x+1}{x^{2}-x}\)
27) \(\quad \frac{x}{x+2}+\frac{x+1}{x^{2}-7 x-18}=\frac{5}{x-9}\)
28) \(\quad \frac{2 a}{a+7}-\frac{a}{a+3}=\frac{5}{a^{2}+10 a+21}\)
29) \(\quad \frac{x-1}{2 x+1}-\frac{2 x-3}{x+3}=\frac{3}{2 x^{2}+7 x+3}\)
30) \(\quad \frac{y}{y+4}+\frac{6}{y+1}=\frac{y^{2}+4}{y^{2}+5 y+4}\)

Addendum - Word Problems
Following are a selection of word problems - some from ancient times, some from the Renaissance and Enlightenment, and some from the 19 th and 20 th century.
1) A teacher agreed to work 9 months for \(\$ 562.50\) and board. At the end of the term, on account of two months absence caused by illness, he received only \(\$ 409.50\) for his seven months work. If the teacher used all nine months of his board during the term, what was his board per month? (American 1892 )

2) A servant is promised \(\$ 100\) plus a cloak as wages for a year. After seven months, he leaves and receives \(\$ 20\) plus the cloak. How much is the cloak worth? (Clavius, German 1608 )

3) The sales tax on garments is \(\frac{1}{20}\) of their value. A certain man buys 42 garments, paying in copper coins. Two garments and 10 copper coins are paid as tax. What is the price of a garment, O learned one? (Ancient India)

4) Two wine merchants enter Paris, one of them with 64 casks of wine, the other with 20 casks (all of the same value). since they do not have enough money to pay the customs duties, the first pays 5 casks of wine and 40 francs, and the second pays 2 casks of wine and receives 40 francs change. What is the price of each cask of wine and what is the duty on each cask? (Chuquet, French 1484 )

5) One of two men had 12 fish and the other had 13 fish, and all of the fish were of the same price. From the first man, a customs agent took away 1 fish and
12 denarii for payment. From the other man he took 2 fish and gave him back 7 denarii as change. Find the customs fee and the price of each fish. (Fibonacci, Italian 1202 )

6) Two traders transporting sheepskins approach their country's border. The first trader has 100 sheepskins and the border guard takes 10 sheepskins plus \(\$ 25\) as a tariff. The second trader has 42 sheepskins and for a tariff, the border guard takes 7 sheepskins but returns \(\$ 14\) change. What is the tariff per sheepskin and what is the value of each sheepskin?

7) Two people have a certain amount of money. The first says to the second, "If you give me 5 denarii, I will have 7 times what you have left." The second says to the first, "If you give me 7 denarii, I will have 5 times what you have left." How much money does each have? Round to the nearest 10 th. (Leonardo, Italian c. 1500 )

8) Two different scenarios from Ancient Greece:
Two friends were walking. One said to the other, "If I had 10 more coins, I would have 3 times as much money as you." The other said, "If I had 10 more coins, I would have five times as much as you." How many coins does each have?

Two friends were walking. One said to the other, "If you give me 10 of your coins, I would have 3 times as much money as you." The other said, "If you give me 10 of your coins, I would have five times as much as you." How many coins does each have?

9) Andy and Betty together have \(\$ 6\) less than Christine. If Betty gives \(\$ 5\) to Andy, then Andy will have half as much as Christine. If, instead, Andy gives \(\$ 5\) to Betty, then Andy will have one-third as much as Betty. How much does each person have to begin with?

10) Three friends \((\mathrm{A}, \mathrm{B} \text { and } \mathrm{C})\) each have a certain amount of money. A says, "I have as much as B plus one-third as much as C." B says, "I have as much as C plus one-third as much as A." Csays, "I have 10 more than one-third of B." How much does each person have? (Ancient Greece)

11) On a test, 39 more pupils passed than failed. On the next test, 7 who passed the first test failed and one-third of those who failed the first test passed the second. As a result, 31 more passed the second test than failed it. What was the record of passing and failing on the first test?

12) At two stations, A and B, six miles apart on the railway, the prices of coal are \(\$ 20\) per ton and \(\$ 24\) per ton respectively. The rates of cartage of coal are \(\$ 2.00\) per ton per mile from A and \(\$ 3.00\) per ton per mile from B. At a certain customer's home, on the railroad from \(\mathrm{A}\) to \(\mathrm{B}\), the cost of coal is the same whether delivered from A or B. Find the distance to this home from A.

13) There were three-fourths as many women as there were men on the train. At the next station six men and eight women got off the train, and twelve men and five women got on. There were then three-fifths as many women as men on the train. How many men and how many women were originally on the train?

14) If a theater could put 5 more seats in a row, it would need 20 rows less, but if each row had 3 fewer seats, it would take 20 rows more to seat the same number. How many people will it seat?

15) An audience of 450 people is seated in rows, with the same number of people in each row. It would take 5 rows less if 3 more people were seated in each row. In how many rows are the people seated?

16) If evergreens are planted 4 feet closer together it will take 44 more trees for a certain piece of road, but if they are planted 6 feet farther apart, it will take 44 fewer trees for the same length of road. How many miles is the piece of road? (use \(5280 \text { feet }=1 \text { mile })\)

17) A movie theater owner found that by raising the price of each ticket by
\(\$ 1.00,200\) fewer people attened and she broke even, but that if she lowered the price by \(\$ 1.00\) per person, 550 people attended and she increased her receipts by
\(\$ 1000 .\) What is the usual rate per person?

18) The brine pipes in an 84 -foot width artificial hockey rink are equally spaced. If the space between each pair of pipes were increased by 1 inch, then 84 fewer lengths of pipe would be needed. What is the distance between the pipes now?

19) A living room shelf is 36 inches long and contains a certain number of books of uniform width. If each book were one-half inch narrower, the shelf would hold six more books. How many books of the wider variety does it hold?

20) I am a brazen lion; my spouts are my two eyes, my mouth and flat of my right foot. My right eye fills a jar in two days, my left eye in three and my foot in four. My mouth is capable of filling it in six hours. Tell me how long all four together will take to fill it. (Ancient Greece)

21) A man wishes to have 500 rubii of grain ground. He goes to a mill that has five stones. The first stone grinds 7 rubii of grain in an hour, the second grinds 5 rubii in an hour, the third 4 rubii in an hour, the fourth grinds 3 rubii per hour and the fifth grinds 1 rubii per hour. In how long will the grain be ground and how much is done by each stone? (Clavius, German 1583 )

22) If two men and three boys can plow an acre in one-sixth of a day, how long would it require three men and two boys to plow it? (Edward Brooks, American 1873 )
23) A cobbler can cut leather for ten pairs of shoes in one day. He can finish 5 pairs of shoes in one day. How many pairs of shoes can he both cut and finish in one day? (Ancient Egypt)

24) Four waterspouts are filling a tank. Of the four spouts, one can fill the tank in one day, the second takes two days, the third takes three days and the fourth takes four days. How long will it take all four spouts working together to fill the tank? (Ancient Greece)

25) If, in one day, a person can make 30 arrows or fletch [put feathers on] 20 arrows, how many arrows can this person both make and fletch in one day? (Ancient China)

26) One military horse and one ordinary horse can pull a load of 40 dan. Two ordinary horses and one inferior horse can pull the same load of 40 dan as can three inferior horses and one military horse. How much can each horse pull individually? (Ancient China)

27) A barrel of water has several holes in it. The first hole empties the full barrel in 3 days. The second hole empties the full barrel in 5 days. The third hole empties the full barrel in 20 hours and the last hole empties the full barrel in 12 hours. With all the holes open, how long will it take to empty the barrel? (Levi ben Gershon, French 1321 )

28) A certain lion could eat a sheep in 4 hours; a leopard could do so in 5 hours; and a bear in 6 hours. How many hours would it take for all three animals to devour a sheep if it were thrown in among them? (Fibonacci, Italian 1202 )

29) Two ships are some distance apart, which journey the first can complete in 5 days and the other in 7 days - it is sought in how many days they will meet if they begin the journey at the same time. (Fibonacci, Italian 1202 )

30) Sarah, alone, can paint the garage in 24 hours, her sister Jenny, alone, can paint the same garage in 12 hours. With the aid of their mother, the three together can paint the garage in 4 hours. How long would it take their mother, working alone, to paint the garage?

31) It required 75 workers 38 days to build an embankment to be used for flood control. Had 18 workers been removed to another job at the very start of operations, how much longer would it have taken to build the embankment?

32) Mark, alone, requires 6 hours to paint a fence; however, his younger brother, who alone could do it in 9 hours, helps him. If they start work at 8: 30 am, at what time should they finish the work?

33) A group decides to build a cabin together. The job can be done by 3 skilled workers in 4 days or by 5 amateurs in 6 days. How long would it take if they all work together?

34) If it requires 18 workers 50 days to build a piece of road, how many days sooner would it be done if 7 more workers were hired at the beginning of operations?

35) A contractor estimated that a certain piece of work would be done by 9 carpenters in 8 hours or by 16 amateurs in 9 hours. The contractor wishes to get the job done as quickly as possible and uses both professional carpenters and amateurs on the same job. Four carpenters and 4 amateurs begin work at 6 am. Allowing 45 minuntes for lunch, at what time should they finish the job?

36) Mrs. Ellis alone can do a piece of work in 6 days. Her oldest daughter takes 2 days longer; her youngest daughter takes twice as long as her mother. How long will it take to complete the job if all three work together?

37) If 5 men and 2 boys work together, a piece of work can be completed in one day and if 3 men and 6 boys work together, it can be completed in one day. How long would it take a boy to do the work alone?

38) A coal company can fill a certain order from one mine in 3 weeks and from a second mine in 5 weeks. How many weeks would be required to fill the order if both mines were used?

39) If 25 skilled workers work for 8 days, they can complete the construction of a concrete dam; 12 skilled workers and 15 untrained workers together can complete the dam in 10 days. How long would it take an untrained worker alone to complete the work on the dam?

40) \(A\) and \(B\) working together can complete a piece of work in 30 days. After they have both worked 18 days, however, \(A\) leaves and \(B\) finishes the work alone in 20 more days. Find the time in which each can do the work alone.

41) A dump cart can haul enough gravel to fill a pit in 6 days. A truck can do the same work in 2 days. How long would it take two dump carts and one truck working together to fill the pit?