What this move does
The hard part of a word problem is the “translation”: turning a paragraph into an equation. The translation needs just one thing, an equal relationship: say the same quantity in two different ways and put an equals sign between them.
In journey problems, the two most useful sentences are:
- Meeting (walking toward each other): the sum of the distances the two travel = the distance between them at the start.
- Catching up (walking the same way): the difference of the distances the two travel = the distance between them at the start.
Draw each person's position as a line on a distance–time graph, and both sentences say the same thing: the two lines cross. Solving a word problem with an equation takes four steps:
- Choose: pick one unknown quantity and call it . Say what its unit is and where it is measured from.
- Write: express all the other quantities in terms of , and use the equal relationship to write an equation.
- Solve the equation.
- Check and answer: check that the solution fits the story, then answer the question that was actually asked.
Predict first
B leaves school traveling at km per hour. hour later, A leaves school to chase B at km per hour. How many hours after A leaves does A catch B?
Let B be caught hours after B leaves. B has gone . A left hour later and has gone . When A catches B, they are in the same place:
But is how long B has been traveling. A has traveled for only hours. (Or think of it this way: when A sets off, B is km ahead, and A closes the gap by km each hour, so it takes hours.)
Work through one
Example. Towns P and Q are km apart. A truck (T) drives from P toward Q at km per hour. hour later, a car (C) drives from Q toward P at km per hour. How many hours after the car leaves do they meet?
Example: when do the truck and the car meet?
Step 1 / 6- Choose
The question asks “how many hours after the car leaves”, so call that . The time on the graph is measured from when the truck leaves, so they meet at .
Same graph, different story
A job takes A days working alone, and B days working alone. How many days does it take them working together?
Think of the whole job as parts: A does parts a day and B does part a day. Let A work upward from part and B work downward from part . The moment they “meet”, the job is exactly finished.
Let them finish together in days. Written with fractions, this is
“Work done = rate × time” is the same relationship as “distance = speed × time”. Mixture problems (amount of salt = amount of solution × concentration) and money problems (total cost = unit price × quantity) look the same too: find the “this × that = the other” relationship, then look for the quantity that stays the same before and after.
When it fails
Break it
Counterexample lab: remove one condition
Conclusion: going the same way, the one behind is sure to catch the one in front ← still true?
B is km ahead of A and travels at km per hour; A travels at km per hour. A closes the gap by km each hour and catches B in hours.
Now you can change A's speed. Set A's speed to or less, and see whether the two lines still cross.
At the same speed, the two lines are parallel and never cross; if A is slower, they drift further and further apart. You can still write the equation , but it simplifies to , which has no solution. The equation is honestly telling you: A can never catch up.
So when you solve a word problem, both “the number you found” and “no solution” need to be read back into the story.
Practice
Two towns are km apart. Cars A and B leave the two towns at the same time and drive toward each other, A at km per hour and B at km per hour. After how many hours do they meet?
A younger brother walks at meters per minute and sets off minutes early. His older brother walks at meters per minute and follows the same road from the same place. How many minutes after the older brother sets off does he catch up?
A job takes A days alone and B days alone. A works alone for days, then the two work together. How many more days does the job take? (You may write a fraction.)
Salt water of concentration and of concentration are mixed to make g of salt water of concentration . How many grams of the salt water are needed?