Sandra was asked her age. She answered:

80 – 40÷10×4

How old is Sandra?

As usual, watch the video for a solution.

What this viral math problem taught me about the order of operations

Or keep reading.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Answer To How Old Is Sandra?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

80 – 40÷10×4 = ?

The correct answer is 64 according to the modern interpretation of the order of operations.

But it is possible Sandra is 79, or some other age, and is just happy to troll the internet. As they say, never ask a woman her age!

With that said, it is important for students to understand the order of operations.

The order of operations

The expression can be simplified by the order of operations, often remembered by the acronyms PEMDAS/BODMAS.

First evaluate Parentheses/Brackets, then evaluate Exponents/Orders, then evaluate Multiplication-Division, and finally evaluate Addition-Subtraction. If two operations of the same precedence appear, evaluate from left to right.

According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right.

First take 40 and divide it by 10, and then multiply by 4. Finally subtract that from 80

80 – 40÷10×4
= 80 – (40÷10)×4
= 80 – 4×4
= 80 – 16
= 64

This is without a doubt the correct answer according to the modern interpretation of the order of operations.

But some people may have learned it a different way.

The other result of 79

Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below). In other words you need to do the multiplications first and then the division.

Under that interpretation:

80 – 40÷10×4
= 80 – 40÷(10×4)
(Important: this is outdated usage!)
= 80 – 40÷40
= 80 – 1
= 79

This gives the result of 79. This is not the correct answer that calculators will evaluate; rather it is what someone might have interpreted the expression according to older usage.

Here’s a textbook from 1969 sent to me by Pete P.

1969 Textbook Mathematics for Management and Finance, with Basic and Modern Algebra 2nd Ed., Stephen P. Shao, Ph. D. Professor of Management and Statistics, School of Business Administration, Old Dominion University.

So if some people learned it one way, and other people learned it another way, is there really any saying what the correct answer is?

Calculators and coding languages are unanimous

If you input this expression into Python, Java, Excel, or any cell phone calculator you will get:

40÷10×4 = 16

If the 2 conventions really were equally valid, then evidently the people who learned 40÷10×4 = 1 were not employed by tech companies that made calculators.

I think this is strong evidence there is a left to right preference, consistent with what is taught in schools today (at least in America). So I would say:

80 – 40÷10×4 = 64

But perhaps Sandra actually is 79 years old, wanted to obfuscate her age (never ask a woman her age!), and was happy to troll the internet with the other convention that she learned in school.

However, I do think it’s important to learn what calculators and programming languages will interpret this expression. The day may come where you have to review someone’s code, and you better know how computers parse this expression!

*Note: I get many, many emails arguing with me about these order of operations problems, and most of the time people have misunderstood my point, not read the post fully, or not read the sources. If you send an email on this problem, I may not have time to reply.

References

Chicago History on X
https://x.com/Chicago_History/status/2052495811259347085

Lennes, N. J. “Discussions: Relating to the Order of Operations in Algebra.” The American Mathematical Monthly 24.2 (1917): 93-95. Web. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents

Christman, John Michael. Shop Mathematics. United States, Macmillan, 1922. Page 4.
https://www.google.com/books/edition/Shop_Mathematics/42wXAAAAYAAJ?hl=en&gbpv=1&dq=%22use+parentheses%22+division+symbol&pg=PA4&printsec=frontcover

As usual, watch the video for a solution.

How Tall Is The Dog Puzzle?

Or keep reading.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
P
U
Z
Z
L
E
.
.
.
.
Answer To How Tall Is The Dog Puzzle?

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Method 1: Algebra

Let p be the height of the post, and d be the height of the dog. From the diagram we have:

200 + d = p
p + d = 300

Substituting the first equation for p into the second gives:

200 + d + d = 300
200 + 2d = 300
2d = 100
d = 50 cm

Method 2: Visual Dog

Overlapping the two images of the dog will eliminate the dog variable to give the diagram:

We now have:

2p = 300 + 200
p = 250

Using the right diagram in the original figure we have:

p + d = 300

Since p = 250, we must have d = 50 cm.

Method 3: Visual Pole

We can alternately overlap the images of the poles to eliminate the pole variable:

We thus have:

d + 200 + d = 300
2d + 200 = 300
2d = 100
d = 50

Reference

https://x.com/codek_tv/status/2037414684484321603

An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the symphony? Let P be number of players and T the time playing.

While it appears to be a flawed problem, it actually has an important purpose.

As usual, watch the video for a solution.

Internet Can’t Believe This Math Problem

Or keep reading.

.
.
.
.
M
I
N
D
.
Y
O
U
R
.
D
E
C
I
S
I
O
N
S
.
.
.
.
Internet Can’t Believe This Math Problem

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

References

Symphony
An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the symphony? Let P be number of players and T the time playing.

There will be students that mindlessly use a proportion. They may think that 1/2 as many players will take 2 times as long, or 80 minutes.

Of course the time of a symphony will be fixed regardless of the players, so the answer would be T = 40.

This was intentionally a trick question from a teacher Claire Longmoor, who included it in a problem set of about proportions.

Trick questions are a guard against excessive rote learning and mindlessly using formulas. Here are some examples of how widespread this problem is.

Age of captain
If a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?

In a study in 1979, French researchers found students would just add the numbers to get 26 + 10 = 36 years old. A similar result happened in China in 2018. Of course the answer is there is not enough information.

Age of shepherd
There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

A 1986 paper in Switzerland found students would manipulate the numbers to find a reasonable answer.

125 + 5 = 130 (too old)
125 – 5 = 125 (too old)
125/5 = 25 (about right)

So they would say the shepherd is 25 years old.

Bus remainder
An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?

A 1983 NAEP assessment in America found students realized they needed to perform a division:

1128/36 = 31 remainder 12

But only 23% of students rounded up to the correct answer of 32 buses. Some replied 31 buses. And a few shockingly wrote 31, remainder 12 buses!

Area of triangle
The hypotenuse of a right-angled triangle is 10 inches, and the altitude dropped onto it is 6 inches. Find the area of the triangle.

The Russian mathematicians Vladimir Arnold joked that American students could determine an answer from formulas. But Russian students could not. Why not?

Full details in a previous post.

Cat and mouse problem
If 6 cats capture 6 mice in 6 minutes, how many are needed to capture 100 mice in 50 minutes? (Lewis Carroll)

This problem comes from the famed author Lewis Carroll of Alice in Wonderland fame, who was also a mathematician (real name Charles Dodgson).

Using the theory of proportions, we would have:

(cats)(time)(rate) = mice
(6 cats)(6 min)(rate) = (6 mice)
rate = (1 mouse)/(6 cat-min)

(cats)(50 min)(1 mouse)/(6 cat-min) = 100 mice
cats = 100 mice(6 cat-min/1 mouse)/(50 min)
cats = 12 cats

But let’s think about scaling our original information.

6 cats capture 6 mice in 6 minutes.

But if we double the number of cats, and multiply the minutes by 8, we would increase the number of mice by 2×8 = 16, so then 16×6 = 96 mice would be captured.

12 cats capture 96 mice in 48 minutes

Until 50 minutes has elapsed, we only have 2 minutes for the 12 cats to capture 4 more mice. Is it possible?

This entirely depends on how the 6 cats are capturing 6 mice in 6 minutes. Carroll goes through 4 possibilities.

As cats operate individually and not in groups, it would be reasonable to think each is working at a rate of:

1 cat capture 1 mouse in 6 minutes

But in this case, none of the 12 cats could possibly capture a mouse in the remaining 2 minutes! In order to surpass 100 mice in 50 minutes, we would need 13 cats. This can be seen by scaling the cats by 13 and the minutes by 8, so the mice caught are 13×8 = 104.

13 cats capture 104 mice in 48 minutes

There are 3 other reasonable interpretations.

6 cats capture 1 mouse in 1 minutes
3 cats capture 1 mouse in 2 minutes
2 cats capture 1 mouse in 3 minutes

In the first 2 cases, having 2 minutes will be enough for the 12 cats to capture an additional 4 mice.

But in the last case, 3 minutes exceed the 2 minutes left, so we would need 2 more cats in the group. That is, 12 + 2 = 14 cats.

Carroll notes this problem illustrates a common issue in proportion questions. At first you might calculate an answer that appears correct, but upon closer inspection of the details it turns out the problem is either impossible or would require additional information to be solved.

It is great the teacher included Beethoven’s 9th symphony as a trick question, as students can fall into the trap of mindlessly applying formulas, and it is a good reminder to think about the question when finding a solution.

References

Symphony
https://twitter.com/dmataconis/status/917496578285490178https://twitter.com/LongmoorClaire/status/918014499071897600
https://time.com/4979608/beethoven-trick-question/
http://mashable.com/2017/10/12/math-problem-out-of-context/#nT07fua4Xkqz
http://www.nottinghampost.com/news/maths-problem-created-nottingham-teacher-623274
https://www.popsugar.com/moms/Impossible-Trick-Math-Question-44141099
https://medium.com/@intersectarian/beethovens-ninth-97ec40ff24f2
https://www.mathnasium.com/math-centers/denville/news/math-teacher-keep-kids-on-their-toes-with-music-question
Age of captain
https://en.wikipedia.org/wiki/Age_of_the_captain
How old shepherd
https://medium.com/student-voices/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff#.xb5jcrgzf
https://www.researchgate.net/publication/331454183_I_added_the_numbers_it’s_math_How_sense-making_in_age_of_the_captain_problems_differs_between_a_mathematics_classroom_and_a_language_classroom
Bus remainder
https://www.inference.org.uk/sanjoy/benezet/rote.html
Triangle
http://imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf
http://math.stackexchange.com/questions/1594740/v-i-arnold-says-russian-students-cant-solve-this-problem-but-american-student
Cat and mouse
https://james.fabpedigree.com/thecats.htm
https://archive.org/details/monthlypacket33unkngoog/page/204/mode/2up?q=lewis