A census taker came to a house where a man lived with
3 daughters. “What are your daughters’ ages?” she asked.
The man replied, “The product of their ages is 36, and
the
sum of their ages is equal to our house number.”
“That’s still not enough information” the census taker said,
after looking
at the number.
“Sorry, I have to go”, the man said, “my oldest daughter
needs help making blueberry pancakes”.
The census taker then promptly wrote down the three daughters’ ages and moved on to the next house. How did she know the ages? What are the ages?
Some mathematics problems are challenging because they just do not seem to provide enough information to the reader to be solved. Often, though, that is because we have not looked carefully enough at the question and thought about what each of the words being used implies. To Re-examine the Problem is to look at each and every word, term, and equation in the statement of the problem and to see if there is more information to be gleaned from them if you look more closely.
In the problem above, at first it seems that much of the information given is completely unhelpful and irrelevant. Although there seems to be insufficient information to determine the daughters’ ages, that should prompt us to re-examine the problem to see if some of that “irrelevant” information can in fact be used productively. While knowing the product of the ages is 36 doesn’t tell us the ages of the daughters, it does limit the possibilities of what the ages could be. What are the possibilities? And although we don’t know the house number, the census taker does, and yet she still cannot say for sure what the ages are. So, which of the possibilities that you listed are consistent with that fact? And while blueberry pancakes are soon to be eaten by the man and his daughter, is that the only piece of additional information you can get from that sentence?
Three squares are placed next to each other as shown. The vertices A, B, and C are collinear. Find the dimensions of the largest square. (Copyright Phillips Exeter Academy.)
Sometimes, even when a problem does not seem to be all that complicated, we still find that there seems to be insufficient information. Often that is because we have not looked fully at what the words in the question imply. For example, although you likely took care to label the lengths of all 12 sides of the 3 squares, did you ask yourself what it means for A, B, and C to be collinear? What do line segments AB and BC thus have in common? It is easy for us to “read past” certain words without fully grasping what they are telling us. Carefully re-examining the problem gives us a chance to really slow down and look, as if with a microscope, at each individual part.
A triangle where each side is an integer has a perimeter of 8. What is its area?
A man was looking at a portrait. Someone asked him, “Whose picture are you looking at?” He replied: “Brothers and sisters have I none, but this man’s father is my father’s son.”
Whose picture was the man looking at?
Which will save you the most amount of gas in a year: changing from a 10 mpg to a 20 mpg car, changing from a 20 mpg to a 40 mpg car, or changing from a 40 mpg to an 80 mpg car?
At 6 o’clock the wall clock struck 6 times. Checking with my watch, I noticed that the time between the first and last strokes was 30 seconds. How long will the clock take to strike 12 at midnight?
If 4 copiers can process 400 sheets of paper in 4 hours, how long would it take 8 copiers to process 800 sheets?
If your answer to #7 was “8 hours”, read the question again, slowly, and think carefully about what the information you are given is actually telling you!
Smallville and Tinytown are 200 miles apart and are connected by a straight railroad track. At 2 o’clock a train leaves Smallville at 50 mph and another train leaves Tinytown at 40 mph. When they eventually meet, which train will be closer to Tinytown?
Take a look at the following proof:
$\begin{array}{l} a = b\\ {a^2} = ba\\ {a^2} - {b^2} = ba - {b^2}\\ {a^2} - {b^2} = b(a - b)\\ (a + b)(a - b) = b(a - b)\\ a + b = b\\ 2b = b\\ 2 = 1 \end{array}$
What went wrong?
In the figure below, can one determine $\angle A + \angle B + \angle C + \angle D$ in terms of angle n? If not, explain why. If so, what does it equal?
Amber bought several identical boxes of CrazyCreme cookies to serve at a big party. She noticed as she was leaving the store that the number of boxes she bought is the same as the number of cents in the cost of a single box. If she spent \$37.44 total, what is the cost of a single box of cookies?
How many integers from 1 to 400, inclusive, are not the square of an integer?
If $0 \le w \le 6$ and $- 2 \le p \le 4$, what is the largest and smallest value $wp$ can have?
Three businessmen — Smith, Robinson, and Jones — all live in the Leeds-Sheffield district. Three railwaymen (a guard, a stoker, and an engineer) of similar names live in the same district. The businessman Robinson and the guard live at Sheffield, the businessman Jones and the stoker live at Leeds, while the businessman Smith and the railway engineer live halfway between Leeds and Sheffield. The guard’s namesake earns \$10,000 per annum, and the engineer earns exactly one-third of the salary of the businessman living nearest to him. Finally, the railwayman Smith beats the stoker at billiards. What is the engineer’s name?
For what values of $x$ does
If $(y-z)^2 < a - b$, is $b - a > 2$?
In a triangle where $\angle A$ is the largest angle,
${(\sin A)^2} + {(\tan A)^2} + {(\cos A)^2} = 2$. What does
A standard deck of playing cards contains 26 black cards and 26 red cards. A deck is randomly divided into two unequal piles, such that the probability of drawing a red card from the smaller pile is 1/3. At the same time, the probability of drawing a black card from the larger pile is 5/14. How many cards are in the larger pile?
In isosceles triangle ABC, $\overline {BC} $ is not quite
twice as long as either of the other two equal sides. Which
is larger, $\sin A$ or
A magician put four yellow, four green, and four red eggs in each of two hats. He called a person from the audience, blindfolded him, and asked him to transfer five eggs from hat 1 to hat 2. The magician then asked the audience to tell the blindfolded assistant how many eggs to return to hat 1 to ensure that in hat 1 there will be at least three eggs of each of the three colors. What would you have told the assistant?
Let X and Y be circles of diameter 2 that are tangent to each other in the plane. How many circles of diameter 6 are in this plane and tangent to X and to Y?