$3$ times a number, plus $4$, is equal to $10$

Question

I was helping my brother with his math homework and there was this question:

$3$ times a number, plus $4$, is equal to $10$. What is that number?

My first thought was that $3x+4 = 10$ and then, solve for $x$. But then, my brother told me maybe it’s $3(x+4)=10$. Now I’m confused too. Which one is right? I think it’s the former given that “plus $4$” was in-between commas but I’m not sure.

Thanks in advance!

Answer 1

If the question is as you typed it, the comma after number indicates you are expected to read it as $3x+4=10$. You multiply the number by $3$, pause, and add $4$. It would be hard to express $3(x+4)=10$ without restructuring the sentence.

The language used indicates this is a low level class. The fact that $3x+4=10$ results in a whole number for $x$ while $3(x+4)=10$ does not argues strongly for $3x+4=10$ being the intended interpretation.

Answer 2

THe important thing is that your brother understands the idea that three times (a number plus four, and (three times a number) plus four, are different concepts and will have different results.

After that it's not a matter of which is right but how well did the text express it and what was the text's intent. In other words it's not a matter of math but of language.

If the question were indeed written as "3 times a number, plus 4, is equal to 10" then the commas and the embedded clause makes it clear that they mean $(3x) + 4 = 10$.

I suppose to express $3(x+4)= 10$ with the same word structure one could write "3, times a number plus 4, is equal to 10" but that's a little awkward. I'd probably say something more explicit such as "3 times the sum of a number plus four, is equal to 10".

A sentence without commas, "3 times a number plus 4 is equal to 10", is ambiguous but I think it is natural to hear think left to right: $3x + 4 = 10$. Also the conventions of math is to simply list as you hear the "3" (that's $3$) "times" (that's $\cdot $) "a number" (that's $x$) "plus" (that's $+$) "4" (that's $4$) "is equal to" (that's $=$) "ten" (that's $10$) so it is $3\cdot x + 4 = 10$.

Likewise "3 plus a number times 4 is equal to 10" without commas is most likely to be interpreted as $3+x\cdot 4 = 10$. And conventions of mathematics and order of operations determine the rest.

Answer 3

Almost certainly the problem is intended to be $3x+4=10$.

But your brother has made an important discovery. Until the sixteenth century, all mathematical textbooks would write out such problems in words, like "3 times a number, plus 4, is equal to 10". As your and your brother have discovered, this can be difficult to interpret correctly, and certainly takes up a lot of space. If simply writing down a mathematical equation is difficult and time consuming and error prone then it makes you less likely to spend time working out how to solve the equation. And if you do find a method of solving a particular type of equation then it is much harder to explain your method to other people.

The adoption of standard mathematical symbols such as $+$ and $=$ and the use of letters like $x$ to stand for unknown quantities was a big step forwards in the development of modern mathematics.