Most word-problem errors are made before any algebra happens, in deciding what the letter stands for and which way round a subtraction goes. The translation itself is a short vocabulary plus two rules: which phrases reverse the order, and which phrases create a bracket.
Core ideas
English is ambiguous and algebra is not, so translating always loses something. The skill is knowing which of the possible readings the sentence actually licenses.
"Let n be the number of quarters" is a definition you can check against later. "Let n be the quarters" is not, because it does not say whether n counts coins or measures money. Half the errors in Linear Equations and Inequalities are made in this sentence and inherited by everything after it.
n + 5 and 5 + n are the same expression, so no phrase involving "sum" can be got wrong by ordering. n − 5 and 5 − n are different, so every phrase involving "less than" or "subtracted from" has to be read carefully.
"Twice the sum of n and 3" is 2(n + 3): the phrase names one quantity and then doubles it. "The sum of twice n and 3" is 2n + 3. Word order is doing the work brackets do in algebra, and it does it less clearly.
The material
Naming the unknown, the vocabulary, the phrases that reverse, and the phrases that bracket.
Before translating anything, write a sentence of the form "let n be the number of ..." or "let c be the cost in dollars of ...". It takes a line and it settles two things that otherwise stay vague: what is being counted or measured, and in what units. The reason is that expressions are checked by substituting, and you cannot check 25n unless you know whether n is a number of quarters or an amount of money. If n is the number of quarters then 25n is the value in cents and the expression is right; if n is already an amount in dollars then 25n is meaningless.
Every line above is a convention of English, not of mathematics. They are worth learning because everyone writing problems uses them the same way, and worth distrusting whenever a sentence is unusual: when the English is genuinely ambiguous, the right move is to say which reading you took, not to guess silently.
Division carries this hazard too: "the quotient of n and 4" is n/4, and so is "4 divided into n", but "n divided into 4" is 4/n — whenever a phrase uses "into", read it twice. Subtraction is where it bites most often, because "less than" and "subtracted from" name the second quantity first:
"The sum of n and 3" packages two things into one quantity. Anything applied to that package applies to all of it, which in algebra means a bracket:
In each pair the words are almost the same and the expressions are not close: at n = 5 they differ by 3 in the first pair and by 18 in the second. The bracket is carrying the meaning, and dropping it does not produce a slightly different answer, it produces a different quantity.
Some phrases describe several unknowns at once, and the point of algebra is that one letter covers them all:
Pick a number, read the English sentence aloud with that number in it, and compare with your expression evaluated there. For "twice the sum of n and 3" at n = 5: the sentence says the sum is 8 and twice that is 16; 2(n + 3) gives 16 and 2n + 3 gives 13. That test rules out 2n + 3. It does not prove 2(n + 3) is right — agreement at one number never does — but a wrong translation almost always fails it.
Interactive
Two readings, tested against each other
Method
The first two steps produce no algebra at all, and skipping them is what goes wrong.
"Let w be the width in centimetres." One line, and every later expression can be checked against it. If the problem has two independent unknowns, it needs two letters.
The length is w + 3; the second integer is n + 1; the number of dimes in a purse of 20 coins holding q quarters is 20 − q. Do this before assembling anything, so that the final expression is built from pieces you have already justified.
Work outward from the innermost quantity. Wherever the sentence says "the sum of" or "the quantity", put a bracket round it, and only then apply whatever is done to it.
Substitute a value and compute the sentence in English alongside your expression. Avoid 0 and 1, which make too many wrong expressions agree with the right one, and then check that the value you picked does separate the two readings in front of you: n = 5 is no use for "five less than n", because n − 5 and 5 − n are both 0 there.
Each translation is followed by the test value that rules out the usual rival reading.
Phrases 2 and 3 use the same seven words in a different order and mean different things. English marks the difference with word order alone; algebra marks it with a bracket, which is one reason the translation is worth doing early rather than carrying the sentence around in your head.
Every test above rules a rival reading out. None of them proves the chosen reading correct, because two different expressions can agree at one value — n − 5 and 5 − n both give 0 at n = 5, which is exactly why n = 5 would have been a bad test for phrase 1.
Now fade the wording: "three times the difference of n and 4". The packaged quantity is the difference, so the expression is 3(n − 4). At n = 6 it gives 6; 3n − 4 gives 14 and reveals that only the first quantity was tripled.
For a taxi fare with a $4 fixed charge and $2.50 per mile, define the unknown with units, write the total for an unknown number of miles, and choose a value that separates it from the common product-of-the-whole-sum error. Then translate "four less than three times the sum of a number and two" and test the nearest rival reading.
Something to change about this lesson? Write it down, tick it off when it is handled, and export the list when you want to act on it. Kept in this browser only: nothing is sent anywhere, and nobody is notified.
No recommendations yet for this lesson.