Word problems: turning a sentence into an equation
Almost nobody fails a word problem on the arithmetic. They fail it before the arithmetic starts, on the sentence - and then spend two minutes doing correct sums that answer a question nobody asked.
There is one move. Decide what the unknown is, write it down as a letter, and only then read the sentence again. Everything else on this page is that move applied to the six sentence patterns the exam actually builds problems from.
Why naming it first is the whole technique
A word problem is a sentence about a quantity nobody has told you. Until you have decided which quantity that is, every phrase in the sentence is ambiguous - and the exam writes them so that a plausible wrong choice is always available. “Four times as old” is a relation between two ages; which of them you call x decides whether the equation you get is easy or awful.
Three passes, in this order. Most wasted time on this section comes from starting the third before finishing the first.
The phrases, and what each one becomes
English used in arithmetic is a small language. These are almost all of it:
Sentence to equation
In the sentence
In the equation
is, was, will be, gives
=
of (after a fraction or percent)
multiply
more than, older than, gained
add
less than, younger than, lost
subtract - and mind the order
times as old / as many as
multiply the smaller one
for every A, B
the ratio B ÷ A
in n years, n years ago
add n to every age in the sentence
The one to watch is less than. “Five less than x” is x − 5, not 5 − x. The reversal is easy to make under time pressure and produces a number that looks like an answer.
Work rate: the translation you can almost see
Start here because there is nothing hidden. A job takes a whole number of days; you are asked what fraction is done after some of them. The whole job is 1, one day is 1 divided by the total, and the answer is that multiplied by the days elapsed.
Work rate · from our bank, and it will generate a fresh one each time you practise
A maintenance crew can repaint the whole building in 17 days. Working at the same rate, what part of the job is finished after 8 days?
A17/8
B8/17
C9/17
D8/25
A uniform rate means each day finishes the same share of the job: one day is 1/17 of it. After 8 days the crew has finished 8/17, which reduces to 8/17. Check the direction before dividing: the part done is always the smaller number over the larger one.
AThis divides the whole by the part. The part finished cannot be larger than the whole job.
CThis is the part of the job still to do, which is what is left rather than what is done.
DThis compares the elapsed days with the total of both figures instead of with the length of the job.
The distractors here are worth reading as a set. They are not random numbers - they are what you get from the plausible wrong translations, which is what makes them useful to study and useless to guess between.
Ratio: keep the two numbers in the order the sentence gave them
“For every 7 harvested, a tenant receives 4” is the fraction 4/7, applied to whatever total the question names. The mistake is not the arithmetic; it is writing 7/4 because 7 came first in the sentence.
Ratio and share
For every 7 sacks of palay harvested, a tenant receives 4. At the same ratio, how many does the tenant receive when 511 are harvested?
A894
B219
C292
D296
The ratio is 4 to 7, so find how many times 7 fits into 511: it fits 73 times. Scale the other side by the same figure: 4 x 73 = 292. Name which side of the ratio the question asks for before you multiply; the other side is always on offer.
AThis uses the ratio upside down, which gives more than the whole quantity.
BThis is what is left after the tenant's share, not the share itself.
DThis adds the ratio's first figure to the answer instead of scaling by it.
Profit and loss: the base is what you paid
Percent gain and percent loss are both measured against the cost, never against the selling price. In the item below, option A is exactly what you get from dividing by the selling price instead - and the rationale under it says so.
Profit and loss
A trader bought a sack of rice for P340 and sold it for P204. What was the percent gain or loss?
A66.7% loss
B40% loss
C40% gain
D80% loss
The difference is P136, and it is measured against the cost: P136 / P340 = 40%. It sold for less than it cost, so that is a loss. Dividing by the selling price instead is the standard error here, and it always produces a believable number - which is why it is on the list.
AThis divides by the selling price. Gain and loss are always measured against the cost, which is P340.
CThe item sold for less than it cost, so this is a loss.
DThis is double the actual rate.
Age: translate each clause separately, then combine
Age problems look harder than they are because two clauses are describing two different moments. Write the present relation, then rewrite every age with the years added, and only then set the two against each other.
Age · two moments, two equations
The head clerk is 32 years older than the messenger. In 8 years, the head clerk will be 3 times as old as the messenger is then. How old is the head clerk now?
A24
B8
C40
D48
Call the younger age x. Today the older is x + 32. In 8 years the younger is x + 8 and the older is x + 32 + 8, and the second is 3 times the first. That gives x = 8, so the older is 40 today. The trap is applying the second condition to today's ages. The two sentences describe different moments, and the whole difficulty of an age problem is keeping them apart.
AThis applies the later multiple to today's younger age. The two conditions describe different moments.
B8 is the younger age. The question asks for the older one.
D48 is the age 8 years from now, not today.
In eight years both people are eight years older. Adding the years to one of them and not the other is the single commonest error on this format, and it produces an answer that is on the list.
Mixture: two things being counted at once
A blend problem is two equations pretending to be one sentence: how much stuff there is, and how much it is worth. Name one quantity x, make the other the total minus x, and write both.
Mixture
A dealer wants 45 sacks of fertiliser worth P45 a sack, blended from stock at P30 and stock at P55. How many sacks of the P30 stock are needed?
A43
B27
C18
D22.5
Let x be the sacks at P30. Then 45 - x are at P55, and the money has to balance: 30x + 55(45 - x) = 45 x 45, which gives x = 18. The instinct to average the two prices is what this format is testing. Averaging only works when the amounts are equal, and here they are exactly what you are being asked to find.
AThis is an average of the two prices, which is not a quantity at all.
B27 is how much of the dearer grade is needed. The question asks for the cheaper one.
DAn even split would only give the blend if the target price were exactly halfway between P30 and P55, and it is not.
Two rates: the one where the wrong unknown costs you the item
Up at one speed, down at another, total time given, asked for the distance. If you name the time as your unknown you get two unknowns and no way forward. Name the distance - it is the same going up and coming down - and the two times are both expressions in it.
Two rates · name the distance, not the time
A technician climbs a tower at 6 feet per minute and descends at 20 feet per minute. The whole trip up and back took 13 minutes. How tall is the tower, in feet?
A85
B130
C78
D60
Write each leg as a time: going up takes d/6 minutes and coming down takes d/20. Their sum is the 13 minutes given, so d(1/6 + 1/20) = 13, and d = 60 feet. Check it: 10 minutes up plus 3 down is 13. Two rates over the same distance never average; the slower leg takes more of the time.
AThis averages the two rates. Time spent at the slower rate is longer, so the two do not weigh equally.
BThis splits the time equally between the two legs, but the legs take different lengths of time.
CThis applies the climbs rate to the whole time, which ignores the return leg.
Note what happens to the average speed here. It is not the average of the two rates - option A above is that average, and it is wrong. Time is distance over speed, so the slow leg takes a larger share of the total time than the fast one and drags the average below the midpoint.
The two mistakes people actually make
Answering a different question. You solved for the daughter’s age and the question asked for the father’s. The number you found is on the option list, because the person setting the item knew you would. The fix costs three seconds: read the last line of the question again before you choose.
Starting the arithmetic during the first read. It feels like progress and it is the reason the sentence gets translated wrongly - you are doing sums while you are supposed to be deciding what the sums are about. Read the whole thing once with your pen down.
Practising it
Volume helps here more than it does anywhere else on the paper, because the point is to recognise the pattern before you have finished reading the sentence, and that is a matter of exposure rather than understanding. Word problems practice draws fresh items in every format above for as long as you keep going - the numbers and the wording change each time, so there is nothing to memorise and no answer key to remember.