How do you translate into mathematical expressions and find the number given Three less than two-thirds of a number is three?

Feb 9, 2017

The mathematical expression is $\frac{2}{3} n - 3 = 3$
The number is $9$

Explanation:

Let the unknown number be shown as $n$. $\frac{2}{3}$ of the number is then $= \frac{2}{3} n$

Three less than $\frac{2}{3} n$ is $= \frac{2}{3} n - 3$

All of this is equal to $3$, so

$\frac{2}{3} n - 3 = 3$

Multiply everything in the equation by three:

$2 n - 9 = 9$

$2 n = 18$

$n = 9$

Check the answer in the new mathematical expression:

$\frac{2}{3} n - 3 = 3$

$\frac{2}{3} \left(9\right) - 3 = 3$

$6 = 3 + 3$

$6 = 6$

Feb 9, 2017

Explanation:

Let's write the sentence down, so we can translate it piece-by-piece into a mathematical equation:

$\text{Three less than two-thirds of a number is three.}$

The first thing to notice is that we can translate the numbers directly:

$\stackrel{3}{\overbrace{\text{Three"" less than " stackrel (2//3) overbrace"two-thirds" " of a number is " stackrel 3 overbrace"three}}} .$

The phrase "a number" refers to our unknown value, because it doesn't specify which number—just a number. We usually choose to represent our unknown number with an $x$ (but you can choose whatever variable you like).

$\stackrel{3}{\overbrace{\text{Three"" less than " stackrel (2//3) overbrace"two-thirds" " of "stackrel x overbrace"a number"" is " stackrel 3 overbrace"three}}} .$

That does it for the values (known and unknown). Now it's time to translate the operations/symbols.

Again, some single words here have direct mathematical translations. The easiest is... well, "is". The word "is" can be replaced with "equals". (Example: if I say "$x \text{ is five}$", you would translate that quite easily as "$x \text{ equals 5}$" ($x = 5$). So "is" becomes "equals" [$=$].)

Similarly, the word "of" becomes multiplication. For example, if I asked you, $\text{What's one-half of 4?}$, you might not realize it, but when you find the answer, you're really solving "one-half times four" $\left(\frac{1}{2} \times 4\right)$, which gives you the answer of 2. So "of" becomes "multiplied by" (or "times").

Using $=$ for "is" and $\times$ for "of", we continue translating:

$\stackrel{3}{\overbrace{\text{Three"" less than " stackrel (2//3) overbrace"two-thirds"" "stackrel xx overbrace"of"" "stackrel x overbrace"a number"" "stackrel = overbrace"is"" "stackrel 3 overbrace"three}}} .$

The only thing left to translate is "less than". Sadly, here is where our word-for-word translation stops. It's not hard to see that "less than" will become subtraction ($-$), but we can't just put a minus sign in there and be done.

Think about it: what's one less than seven? Six, right? But you didn't find that by subtracting $1 - 7$. You did it by subtracting $7 - 1$. In other words, when "less than" appears between two terms, we need to swap the order of the two terms, and then put a minus sign between them. In other words, "$a$ less than $b$" becomes "$b$ minus $a$".

So we need to swap the two terms on either side of the "less than". That will be the "3" on the left, and the "$\frac{2}{3}$ of $x$" on the other, because we're subtracting 3 from "two-thirds of $x$", not just from two-thirds. (Remember: $+ \text{ and } -$ separate terms, while $\times \text{ and } \div$ create them.)

After turning "less than" into "minus" and swapping the order of the associated terms, we get

$\stackrel{2 / 3}{\overbrace{\text{Two-thirds"" "stackrel xx overbrace"of"" "stackrel x overbrace"a number"", "stackrel - overbrace"minus"" "stackrel 3 overbrace"three", stackrel = overbrace"is"" "stackrel 3 overbrace"three}}} .$

And there it is—the translated equation!

$\frac{2}{3} \times x - 3 = 3 \text{, }$ or $\text{ } \frac{2}{3} x - 3 = 3$.

From here, the solution is found by adding 3 to both sides:

$\frac{2}{3} x - \cancel{3} + \cancel{\textcolor{red}{3}} = 3 + \textcolor{red}{3}$

$\textcolor{w h i t e}{\cancel{3} + \cancel{3} -} \frac{2}{3} x = 6$

then multiplying both sides by the reciprocal of $\frac{2}{3}$:

$\cancel{\textcolor{red}{\frac{3}{2}}} \times \cancel{\frac{2}{3}} x = \textcolor{red}{\frac{3}{2}} \times 6$

$\textcolor{w h i t e}{\cancel{\frac{3}{2}} \times \cancel{\frac{2}{3}}} x = \frac{3}{2} \times 6 = \frac{3 \times 6}{2} = \frac{18}{2} = 9$

Thus, after all that, we've found our number: it is 9.

Let's verify it too: what is three less than two-thirds of nine?

$\textcolor{w h i t e}{=}$"$\text{3 less than "2/3" of 9}$"

$=$"$3 \text{ less than 6}$"

$=$"$3$",

which is what we were hoping for.