Equation Word Problems

A word problem describes a situation in words and asks for a number: a price, an age, a distance. Algebra solves it in two stages. First you translate the words into an equation, then you solve the equation. Solving is covered in Solving Linear Equations. This page is about translating, which is where most of the difficulty lies.

The idea that makes algebra work is simple: the number you are looking for gets a name, usually $x$, and you write down what the problem says about it as if it were already known. The equation then does the rest.

On this page: How to solve a word problem · Words into algebra · Number problems · Parts of a whole · Ages · Sharing and comparing · Ratios · Motion · Negative answers · Common mistakes

How to Solve a Word Problem

1. Choose the unknown. Decide what you are looking for and give it a letter: "Let $x$ be the price of the watch in dollars." Write this sentence down.
2. Write the other amounts with $x$. Every amount in the problem should be either a number or an expression with $x$.
3. Write the equation. Find two expressions that the problem says are equal, and put an equals sign between them.
4. Solve the equation.
5. Check and answer. Check the answer against the words of the problem, not just your equation, and answer the question that was asked, with units.

Example. A man was asked how much he paid for his watch. He answered: "If you multiply the price by 4, add 70 and then subtract 50, you get 220 dollars." What was the price?

Solution: Let the price be $x$ dollars, and translate the answer piece by piece:

Words	Algebra
the price	$x$
multiply the price by 4	${4x}$
add 70	${4x+70}$
then subtract 50	${4x+70-50}$
you get 220	${4x+70-50=220}$

Now solve the equation:

${{4x+20}={220}}$, so ${4x=200}$ and ${x=50}$

Check with the words: ${4\cdot 50=200}$, then ${200+70=270}$, then ${270-50=220}$ ✓

Answer: the watch cost $\$50$.

Translating Words into Algebra

These phrases come up again and again. Here $x$ is the unknown number.

Words	Algebra
7 more than a number, a number increased by 7	${x+7}$
7 less than a number, a number decreased by 7	${x-7}$
7 minus a number	${7-x}$
twice a number, double a number	${2x}$
3 times a number, plus 10	${3x+10}$
3 times the sum of a number and 10	${3(x+10)}$
half of a number	${\frac{x}{2}}$
two thirds of a number	${\frac{2x}{3}}$
a number divided by 12	${\frac{x}{12}}$
is, equals, gives, will be, amounts to	$=$
$A$ is 20 more than $B$, $A$ exceeds $B$ by 20	${A=B+20}$
three consecutive whole numbers	$x$, ${x+1}$, ${x+2}$

"7 less than a number" is ${x-7}$, not ${7-x}$: you start with the number and take 7 away. With subtraction and division the order matters, so read these phrases carefully.

Several Amounts, One Letter

Many problems talk about two or more unknown amounts. If you know how they are related, you can write all of them with the same letter:

The problem says	Write the amounts as
two numbers add up to 48	$x$ and ${48-x}$
two numbers differ by 40	$x$ and ${x+40}$
Ben has three times as much as Amy	$x$ for Amy and ${3x}$ for Ben
two amounts are in the ratio ${9:7}$	${9x}$ and ${7x}$

Give the letter to the amount that the others are described by, which is usually the smallest. This avoids fractions: calling Amy's amount $x$ makes Ben's ${3x}$, while calling Ben's amount $x$ would make Amy's ${\frac{x}{3}}$.

Number Problems

In these problems the unknown is just a number, and the sentence tells you what is done to it.

Example 1. If half of a number is added to the number and 20 is subtracted from the sum, the result is a quarter of the number. Find the number.

Solution: Let the number be $x$. Half of it is ${\frac{x}{2}}$ and a quarter of it is ${\frac{x}{4}}$, so

${{x+\frac{x}{2}-20}={\frac{x}{4}}}$

Multiply both sides by 4 to clear the fractions:

${{4x+2x-80}={x}}$, so ${5x=80}$ and ${x=16}$

Check: ${16+8-20=4}$, and a quarter of 16 is 4 ✓

Example 2. Split 48 into two parts so that the smaller part divided by 4 plus the larger part divided by 6 equals 9.

Solution: Let the smaller part be $x$. The two parts add up to 48, so the larger part is ${48-x}$:

${{\frac{x}{4}+\frac{48-x}{6}}={9}}$

Multiply by 12:

${{3x+2(48-x)}={108}}$

${{3x+96-2x}={108}}$, so ${x=12}$

The parts are 12 and 36. Check: ${{\frac{12}{4}+\frac{36}{6}}={3+6}={9}}$ ✓

Try these. Open a problem to see its solution.

1. If a number is divided by 12, the result, the number and 12 add up to 64. Find the number.

Let the number be $x$: ${{\frac{x}{12}+x+12}={64}}$. Multiply by 12: ${{x+12x+144}={768}}$, so ${13x=624}$ and ${x=48}$.

Check: ${4+48+12=64}$ ✓

2. A number is as much less than 500 as a fifth of it is greater than 40. Find the number.

Let the number be $x$. It is ${500-x}$ less than 500, and its fifth is ${\frac{x}{5}-40}$ greater than 40, so

${{500-x}={\frac{x}{5}-40}}$

Multiply by 5: ${{2{,}500-5x}={x-200}}$, so ${2{,}700=6x}$ and ${x=450}$.

Check: ${500-450=50}$ and ${90-40=50}$ ✓

3. A sixth of a number is 20 more than an eighth of it. Find the number.

${{\frac{x}{6}-\frac{x}{8}}={20}}$. Multiply by 24: ${4x-3x=480}$, so ${x=480}$.

Check: ${80-60=20}$ ✓

4. A quarter of a number is 96 more than a fifth of it. Find the number.

${{\frac{x}{4}-\frac{x}{5}}={96}}$. Multiply by 20: ${5x-4x=1{,}920}$, so ${x=1{,}920}$.

Check: ${480-384=96}$ ✓

5. A third, a quarter and two sevenths of a number add up to 73. Find the number.

${{\frac{x}{3}+\frac{x}{4}+\frac{2x}{7}}={73}}$. Multiply by 84: ${28x+21x+24x}={73\cdot 84}$, so ${73x=73\cdot 84}$ and ${x=84}$.

Check: ${28+21+24=73}$ ✓

6. If 10 is added to a number, three fifths of the sum is 66. Find the number.

${{\frac{3}{5}(x+10)}={66}}$. Multiply by 5 and divide by 3: ${x+10=110}$, so ${x=100}$.

Check: ${\frac{3}{5}\cdot 110=66}$ ✓

7. If 720 is added to a number and the sum is divided by 125, the result is the same as 7,392 divided by 462. Find the number.

First work out the known side: ${7{,}392\div 462=16}$. Then ${{\frac{x+720}{125}}={16}}$, so ${x+720=2{,}000}$ and ${x=1{,}280}$.

Check: ${\frac{2{,}000}{125}=16}$ ✓

8. Split 68 into two parts so that 84 minus the larger part is three times as much as 40 minus the smaller part.

Let the larger part be $x$; the smaller part is ${68-x}$:

${84-x}={3\left(40-(68-x)\right)}$

${{84-x}={3x-84}}$, so ${168=4x}$ and ${x=42}$

The parts are 42 and 26. Check: ${84-42=42}$ and ${3(40-26)=42}$ ✓

9. Split 36 into three parts so that half of the first, a third of the second and a quarter of the third are all equal.

Call the equal value $x$. Then the parts are ${2x}$, ${3x}$ and ${4x}$, so ${2x+3x+4x=36}$, ${9x=36}$ and ${x=4}$.

The parts are 8, 12 and 16. Check: half of 8, a third of 12 and a quarter of 16 are all 4 ✓

Parts of a Whole

When a problem describes fractions of a total, let $x$ be the total. The parts, written with $x$, add up to $x$.

Example. Three siblings share an inheritance. The first gets $\$1{,}000$ less than half of it, the second $\$800$ less than a third of it, and the third $\$600$ less than a quarter of it. How much is the inheritance?

Solution: Let the inheritance be $x$ dollars. The three shares are

${\frac{x}{2}-1{,}000}$, ${\frac{x}{3}-800}$ and ${\frac{x}{4}-600}$

and together they make up the whole inheritance:

${\frac{x}{2}-1{,}000}+{\frac{x}{3}-800}+{\frac{x}{4}-600}={x}$

Multiply by 12:

${6x-12{,}000}+{4x-9{,}600}+{3x-7{,}200}={12x}$

${{13x-28{,}800}={12x}}$, so ${x=28{,}800}$

The inheritance is $\$28{,}800$, and the shares are $\$13{,}400$, $\$8{,}800$ and $\$6{,}600$. Check: ${13{,}400+8{,}800+6{,}600}={28{,}800}$ ✓

Try these.

10. A third, a quarter and a fifth of a sum of money add up to $\$94$. What is the sum?

${{\frac{x}{3}+\frac{x}{4}+\frac{x}{5}}={94}}$. Multiply by 60: ${20x+15x+12x}={5{,}640}$, so ${47x=5{,}640}$ and ${x=120}$.

The sum is $\$120$. Check: ${40+30+24=94}$ ✓

11. Mark has lived a third of his life in England, a quarter of it in Scotland and the remaining 20 years in the United States. How old is he?

Let Mark be $x$ years old: ${{\frac{x}{3}+\frac{x}{4}+20}={x}}$. Multiply by 12: ${{4x+3x+240}={12x}}$, so ${240=5x}$ and ${x=48}$.

Mark is 48. Check: ${16+12+20=48}$ ✓

12. A post stands in a pond. A fifth of its length is in the ground, three sevenths are in the water, and 13 feet stick out above the water. How long is the post?

Let the post be $x$ feet long: ${{\frac{x}{5}+\frac{3x}{7}+13}={x}}$. Multiply by 35: ${{7x+15x+455}={35x}}$, so ${455=13x}$ and ${x=35}$.

The post is 35 feet long. Check: ${7+15+13=35}$ ✓

13. In an orchard, three quarters of the trees are apple trees, a tenth are pear trees and the rest are peach trees. There are 20 more peach trees than an eighth of all the trees. How many trees are in the orchard?

Let there be $x$ trees. The peach trees are ${x-\frac{3x}{4}-\frac{x}{10}}={\frac{3x}{20}}$, so

${{\frac{3x}{20}}={\frac{x}{8}+20}}$

Multiply by 40: ${6x=5x+800}$, so ${x=800}$.

Check: there are 600 apple, 80 pear and 120 peach trees, and ${\frac{800}{8}+20=120}$ ✓

14. Four children share an inheritance. The first gets $\$200$ more than a quarter of it, the second $\$340$ more than a fifth, the third $\$300$ more than a sixth, and the fourth $\$400$ more than an eighth. How much is the inheritance?

Let it be $x$ dollars. The four shares make up the whole:

${\frac{x}{4}+200}+{\frac{x}{5}+340}+{\frac{x}{6}+300}+{\frac{x}{8}+400}={x}$

Multiply by 120:

${30x+24{,}000}+{24x+40{,}800}+{20x+36{,}000}+{15x+48{,}000}={120x}$

${{89x+148{,}800}={120x}}$, so ${148{,}800=31x}$ and ${x=4{,}800}$

The inheritance is $\$4{,}800$. The shares are $\$1{,}400$, $\$1{,}300$, $\$1{,}100$ and $\$1{,}000$, and they add up to $\$4{,}800$ ✓

15. After spending $\$100$ more than a third of her monthly income, Lisa still had $\$35$ more than half of it. What is her monthly income?

Let the income be $x$ dollars. Lisa spent ${\frac{x}{3}+100}$, so

${x-\left(\frac{x}{3}+100\right)}={\frac{x}{2}+35}$

Multiply by 6: ${{6x-2x-600}={3x+210}}$, so ${x=810}$.

Her income is $\$810$. Check: she spent ${270+100=370}$ and had ${810-370=440}$ left, which is ${405+35}$ ✓

16. A full water tank lost a third of its water through a leak. Then 21 liters were drawn off, and the tank was half full. How much does the tank hold?

Let it hold $x$ liters: ${{x-\frac{x}{3}-21}={\frac{x}{2}}}$. Multiply by 6: ${{6x-2x-126}={3x}}$, so ${x=126}$.

The tank holds 126 liters. Check: ${126-42-21=63}$, which is half of 126 ✓

17. At a school, half of the students plus 36 walk to school, an eighth of them plus 6 come by bike, and the rest, who are a fifth of all the students, come by bus. How many students are there?

Let there be $x$ students. The three groups make up the whole school:

${\frac{x}{2}+36}+{\frac{x}{8}+6}+{\frac{x}{5}}={x}$

Multiply by 40: ${20x+1{,}440}+{5x+240}+{8x}={40x}$, so ${{33x+1{,}680}={40x}}$, ${1{,}680=7x}$ and ${x=240}$.

There are 240 students. Check: 156 walk, 36 come by bike and 48 by bus, and ${156+36+48=240}$ ✓

18. A bag of trail mix holds nuts, raisins and chocolate chips. The nuts weigh 10 g more than two thirds of the whole bag, the raisins 4.5 g less than a sixth of the whole bag, and the chocolate chips 2 g less than a seventh of the nuts. How much does the bag weigh?

Let the bag weigh $x$ grams. The nuts weigh ${\frac{2x}{3}+10}$, the raisins ${\frac{x}{6}-4.5}$ and the chocolate chips

${\frac{1}{7}\left(\frac{2x}{3}+10\right)-2}={\frac{2x}{21}+\frac{10}{7}-2}$

The three parts make up the whole bag:

${\frac{2x}{3}+10}+{\frac{x}{6}-4.5}+{\frac{2x}{21}+\frac{10}{7}-2}={x}$

Multiply by 42:

${28x+420}+{7x-189}+{4x+60-84}={42x}$

${{39x+207}={42x}}$, so ${207=3x}$ and ${x=69}$

The bag weighs 69 g: 56 g of nuts, 7 g of raisins and 6 g of chocolate chips. Check: ${56+7+6=69}$ ✓

Age Problems

Let $x$ be one person's age, usually the youngest, and write the other ages from it. Everyone gets older at the same rate: in 5 years, each age grows by 5.

Example. Six siblings were born 4 years apart, and the oldest is three times as old as the youngest. How old is each?

Solution: Let the youngest be $x$ years old. The ages are

$x$, ${x+4}$, ${x+8}$, ${x+12}$, ${x+16}$, ${x+20}$

The oldest is three times as old as the youngest:

${{x+20}={3x}}$, so ${20=2x}$ and ${x=10}$

The siblings are 10, 14, 18, 22, 26 and 30 years old. Check: ${30=3\cdot 10}$ ✓

Try these.

19. Grandpa is twice as old as Dad, and Dad is three times as old as Sam. Together they are 140 years old. How old is each?

Let Sam be $x$ years old. Then Dad is ${3x}$ and Grandpa is ${2\cdot 3x=6x}$:

${{x+3x+6x}={140}}$, so ${10x=140}$ and ${x=14}$

Sam is 14, Dad is 42 and Grandpa is 84. Check: ${14+42+84=140}$ ✓

20. A mother is 30 years older than her daughter. In 5 years she will be three times as old as her daughter. How old are they now?

Let the daughter be $x$ years old; the mother is ${x+30}$. In 5 years they will be ${x+5}$ and ${x+35}$:

${{x+35}={3(x+5)}}$

${{x+35}={3x+15}}$, so ${20=2x}$ and ${x=10}$

The daughter is 10 and the mother is 40. Check: in 5 years they will be 15 and 45, and ${45=3\cdot 15}$ ✓

Sharing and Comparing Amounts

When amounts are described by comparing them with each other ("15 more than", "as much as the other two together"), give the letter to the amount the others are built from, then write the rest one by one.

Example 1. For a school party, 146 liters of fruit punch are made from orange juice, apple juice and water. There are 15 liters more apple juice than orange juice, and as much water as both juices together. How much of each is there?

Solution: Let there be $x$ liters of orange juice. Then there are ${x+15}$ liters of apple juice, and the water is

${{x+(x+15)}={2x+15}}$

All three add up to 146 liters:

${x+(x+15)+(2x+15)}={146}$

${{4x+30}={146}}$, so ${4x=116}$ and ${x=29}$

There are 29 liters of orange juice, 44 liters of apple juice and 73 liters of water. Check: ${29+44+73=146}$, and ${29+44=73}$ ✓

Example 2. A grandmother shared some money among her four grandchildren. The third got $\$9$ more than the fourth, the second $\$12$ more than the third, and the first $\$18$ more than the second. In all she gave away $\$6$ more than 7 times what the fourth got. How much money did she share?

Solution: Let the fourth grandchild get $x$ dollars. Then the third got ${x+9}$, the second ${x+21}$ and the first ${x+39}$, which is ${4x+69}$ in all. The problem describes the total in a second way, as ${7x+6}$. The two expressions are equal:

${{4x+69}={7x+6}}$, so ${63=3x}$ and ${x=21}$

The grandchildren got $\$60$, $\$42$, $\$30$ and $\$21$, so she shared $\$153$. Check: ${7\cdot 21+6=153}$ ✓

Try these.

21. Four friends bought a sailboat for $\$4{,}755$. Ben paid three times as much as Amy, Chris paid as much as Amy and Ben together, and Dana paid as much as Ben and Chris together. How much did each pay?

Let Amy pay $x$ dollars. Then Ben paid ${3x}$, Chris ${x+3x=4x}$ and Dana ${3x+4x=7x}$:

${{x+3x+4x+7x}={4{,}755}}$, so ${15x=4{,}755}$ and ${x=317}$

Amy paid $\$317$, Ben $\$951$, Chris $\$1{,}268$ and Dana $\$2{,}219$. Check: ${317+951+1{,}268+2{,}219}={4{,}755}$ ✓

22. Jake bought a bike, a helmet and a lock for $\$360$. The helmet cost twice as much as the lock, and the bike cost twice as much as the helmet and the lock together. Find each price.

Let the lock cost $x$ dollars. The helmet cost ${2x}$, and the bike ${2(x+2x)=6x}$:

${{x+2x+6x}={360}}$, so ${9x=360}$ and ${x=40}$

The lock cost $\$40$, the helmet $\$80$ and the bike $\$240$. Check: ${40+80+240=360}$ ✓

23. A farmer had two flocks of sheep of the same size. After 39 sheep were sold from the first flock and 93 from the second, the first flock had twice as many sheep as the second. How many sheep were in each flock at first?

Let each flock have $x$ sheep:

${{x-39}={2(x-93)}}$

${{x-39}={2x-186}}$, so ${x=147}$

Each flock had 147 sheep. Check: ${147-39=108}$, ${147-93=54}$, and ${108=2\cdot 54}$ ✓

24. The first of two water containers holds three times as much water as the second. After 4 liters are poured out of each, the first holds four times as much as the second. How much water was in each at first?

Let the second container hold $x$ liters; the first holds ${3x}$:

${{3x-4}={4(x-4)}}$

${{3x-4}={4x-16}}$, so ${x=12}$

The containers held 36 and 12 liters. Check: after pouring they hold 32 and 8 liters, and ${32=4\cdot 8}$ ✓

25. Leo bought some paint for $\$94$. He used 7 liters and sold a quarter of the rest to a neighbor for $\$20$, at the price per liter he had paid. How many liters did he buy?

Let Leo buy $x$ liters. One liter cost ${\frac{94}{x}}$ dollars, and he sold ${\frac{x-7}{4}}$ liters for 20 dollars:

${{\frac{x-7}{4}\cdot\frac{94}{x}}={20}}$

Multiply by ${4x}$: ${{94(x-7)}={80x}}$, so ${{94x-658}={80x}}$, ${14x=658}$ and ${x=47}$.

He bought 47 liters, at $\$2$ a liter. Check: he sold ${\frac{47-7}{4}=10}$ liters for ${10\cdot 2=20}$ dollars ✓

26. A shop owner takes $\$50$ a year out of the business to live on. At the end of each year, after the $\$50$ is taken out, the money left in the business grows by a third. After three years the money has doubled. How much did the business start with?

Let the business start with $x$ dollars. Each year, 50 is taken out and the rest is multiplied by ${\frac{4}{3}}$:

after 1 year: ${\frac{4}{3}(x-50)}={\frac{4x-200}{3}}$

after 2 years: ${\frac{4}{3}\left(\frac{4x-200}{3}-50\right)}={\frac{16x-1{,}400}{9}}$

after 3 years: ${\frac{4}{3}\left(\frac{16x-1{,}400}{9}-50\right)}={\frac{64x-7{,}400}{27}}$

This is twice the starting amount: ${{\frac{64x-7{,}400}{27}}={2x}}$, so ${{64x-7{,}400}={54x}}$, ${10x=7{,}400}$ and ${x=740}$.

The business started with $\$740$. Check year by year:

${740-50=690}$ and ${690\cdot\frac{4}{3}=920}$

${920-50=870}$ and ${870\cdot\frac{4}{3}=1{,}160}$

${1{,}160-50=1{,}110}$ and ${1{,}110\cdot\frac{4}{3}=1{,}480}$, which is ${2\cdot 740}$ ✓

Ratio Problems

If two amounts are in the ratio ${a:b}$, write them as ${ax}$ and ${bx}$, where $x$ is the size of one part. If the problem gives a ratio between two expressions, write it as an equation between two fractions and cross-multiply. The ratio and proportion lesson explains both.

Example 1. Divide a prize of $\$2{,}000$ into two parts in the ratio ${9:7}$.

Solution: Let the parts be ${9x}$ and ${7x}$ dollars:

${{9x+7x}={2{,}000}}$, so ${16x=2{,}000}$ and ${x=125}$

The parts are ${9\cdot 125=1{,}125}$ and ${7\cdot 125=875}$ dollars. Check: ${1{,}125+875=2{,}000}$ ✓

Example 2. Two numbers are in the ratio ${2:3}$. If 4 is added to each of them, the ratio becomes ${5:7}$. Find the numbers.

Solution: Let the numbers be ${2x}$ and ${3x}$. After 4 is added to each,

${{\frac{2x+4}{3x+4}}={\frac{5}{7}}}$

Cross-multiply:

${{7(2x+4)}={5(3x+4)}}$

${{14x+28}={15x+20}}$, so ${x=8}$

The numbers are 16 and 24. Check: ${{\frac{16+4}{24+4}}={\frac{20}{28}}={\frac{5}{7}}}$ ✓

Try these.

27. Two numbers differ by 40 and are in the ratio ${6:5}$. Find them.

Let the numbers be ${6x}$ and ${5x}$: ${6x-5x=40}$, so ${x=40}$.

The numbers are 240 and 200. Check: ${240-200=40}$ and ${\frac{240}{200}=\frac{6}{5}}$ ✓

28. The ratio of a number to 12 more than three times the number is ${2:9}$. Find the number.

${{\frac{x}{3x+12}}={\frac{2}{9}}}$. Cross-multiply: ${{9x}={2(3x+12)}}$, so ${9x=6x+24}$, ${3x=24}$ and ${x=8}$.

Check: ${{\frac{8}{3\cdot 8+12}}={\frac{8}{36}}={\frac{2}{9}}}$ ✓

29. Which number, added to both 36 and 52, makes the two sums in the ratio ${3:4}$?

${{\frac{36+x}{52+x}}={\frac{3}{4}}}$. Cross-multiply: ${{4(36+x)}={3(52+x)}}$, so ${{144+4x}={156+3x}}$ and ${x=12}$.

Check: ${\frac{48}{64}=\frac{3}{4}}$ ✓

30. Split 49 into two parts so that the larger part plus 6 and the smaller part minus 11 are in the ratio ${9:2}$.

Let the larger part be $x$. The smaller part is ${49-x}$, and the smaller part minus 11 is ${38-x}$:

${{\frac{x+6}{38-x}}={\frac{9}{2}}}$

${{2(x+6)}={9(38-x)}}$

${{2x+12}={342-9x}}$, so ${11x=330}$ and ${x=30}$

The parts are 30 and 19. Check: ${{\frac{30+6}{19-11}}={\frac{36}{8}}={\frac{9}{2}}}$ ✓

31. Two pieces of fabric with the same price per meter cost $\$50$ and $\$65$. If each piece were 10 meters longer, their lengths would be in the ratio ${5:6}$. How long is each piece?

The price per meter is the same, so the lengths are in the same ratio as the prices, ${{50:65}={10:13}}$. Let the lengths be ${10x}$ and ${13x}$ meters:

${{\frac{10x+10}{13x+10}}={\frac{5}{6}}}$

${6(10x+10)}={5(13x+10)}$

${{60x+60}={65x+50}}$, so ${10=5x}$ and ${x=2}$

The pieces are 20 m and 26 m long. Check: ${{\frac{20+10}{26+10}}={\frac{30}{36}}={\frac{5}{6}}}$ ✓

32. Four towns A, B, C and D lie in that order along a straight road, and A is 34 miles from D. The distances AB and CD are in the ratio ${2:3}$, and a quarter of AB plus half of CD is three times BC. Find AB, BC and CD.

Let ${AB=2x}$ and ${CD=3x}$. Then ${BC=34-5x}$, and

${{\frac{2x}{4}+\frac{3x}{2}}={3(34-5x)}}$

${{2x}={102-15x}}$, so ${17x=102}$ and ${x=6}$

AB is 12 miles, BC is 4 miles and CD is 18 miles. Check: ${12+4+18=34}$, and ${3+9=12}$, which is ${3\cdot 4}$ ✓

Motion Problems

Motion problems use one formula: distance = speed × time. A small table with a row for each traveler helps you write the distances with the unknown.

Example. A truck leaves a warehouse and drives at 60 km/h. One hour later a car leaves the same warehouse and follows it at 75 km/h. How long does the car take to catch up?

Solution: Let the car drive for $t$ hours. The truck started an hour earlier, so it has been driving for ${t+1}$ hours.

	Speed (km/h)	Time (h)	Distance (km)
Truck	60	${t+1}$	${60(t+1)}$
Car	75	$t$	${75t}$

When the car catches up, both have covered the same distance:

${{75t}={60(t+1)}}$

${{75t}={60t+60}}$, so ${15t=60}$ and ${t=4}$

The car catches up after 4 hours, 300 km from the warehouse. Check: ${75\cdot 4=300}$ and ${60\cdot 5=300}$ ✓

Try these.

33. A group of hikers sets off along a long trail, walking 20 km a day. Five days later a second group starts from the same place, walking 25 km a day. After how many days does the second group catch up?

Let the second group walk for $t$ days; by then the first group has walked for ${t+5}$ days:

${{25t}={20(t+5)}}$

${{25t}={20t+100}}$, so ${5t=100}$ and ${t=20}$

The second group catches up after 20 days, 500 km along the trail. Check: ${25\cdot 20=500}$ and ${20\cdot 25=500}$ ✓

34. A ship and a boat are going down a river. When the ship passes under a bridge, the boat is 13 miles farther downstream. The ship covers 5 miles while the boat covers 3. How far below the bridge does the ship catch up with the boat?

Let them meet $d$ miles below the bridge. From the bridge, the ship covers $d$ miles and the boat ${d-13}$ miles in the same time. The ship's speed is to the boat's as 5 to 3, so

${{\frac{d}{5}}={\frac{d-13}{3}}}$

${{3d}={5d-65}}$, so ${65=2d}$ and ${d=32.5}$

The ship catches up 32.5 miles below the bridge. Check: by then the boat has covered ${32.5-13=19.5}$ miles, and ${\frac{32.5}{19.5}=\frac{5}{3}}$ ✓

Negative and Zero Answers

When a problem has amounts in opposite directions, such as profit and loss or north and south, give one direction the plus sign and the other the minus sign. A negative answer then means the opposite direction.

Example 1. A shop made a profit or a loss in January. In February it made a profit of $\$350$, and in March a loss of $\$60$. Over the three months it made a profit of $\$200$. What happened in January?

Solution: Count profits as positive and losses as negative, and let $x$ be the January result:

${{x+350-60}={200}}$, so ${x=-90}$

The minus sign means a loss: the shop lost $\$90$ in January. Check: ${-90+350-60=200}$ ✓

Example 2. A ship sails 4° north, then 13° south, then 17° north, then 19° south, and ends up at 11° south latitude. Where did it start?

Solution: Count north as positive and south as negative, and let $x$ be the starting latitude:

${{x+4-13+17-19}={-11}}$

${{x-11}={-11}}$, so ${x=0}$

Latitude 0 is the equator: the ship started on the equator.

If the answer can't be right, such as a negative number of people, a fraction of a sheep or an age of 300, don't just write it down. Go back to the equation: usually a phrase was translated wrongly.

Common Mistakes

Wrong	Right
✗ "7 less than $x$" written as ${7-x}$	✓ ${x-7}$
✗ Two numbers that add up to 48 written as $x$ and ${x+48}$	✓ $x$ and ${48-x}$
✗ A speed in km/h multiplied by a time in minutes	✓ Use the same units everywhere: 20 minutes is ${\frac{1}{3}}$ of an hour
✗ Stopping at ${x=125}$ when the question asks for the parts ${9x}$ and ${7x}$	✓ Answer the question that was asked: the parts are 1,125 and 875
✗ Checking the answer only in your own equation	✓ Check it against the words: if the equation is wrong, the answer still fits it