Question

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

Answer 1

The mathematical expression is `2/3n - 3 = 3`
The number is `9`

Explanation:

Let the unknown number be shown as `n`. `2/3` of the number is then `= 2/3n`

Three less than `2/3n` is `= 2/3n - 3`

All of this is equal to `3`, so

`2/3n - 3 = 3`

Multiply everything in the equation by three:

`2n -9 = 9`

`2n = 18`

`n = 9`

Check the answer in the new mathematical expression:

`2/3n - 3 = 3`

`2/3(9) - 3 = 3`

`6 = 3 + 3`

`6 = 6`

Answer 2

See explanation. (Warning: detailed answer ahead!)

Explanation:

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

`"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"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"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" is five"`", you would translate that quite easily as "`x" equals 5"`" (`x=5`). So "is" becomes "equals" [`=`].)

Similarly, the word "of" becomes multiplication. For example, if I asked you, `"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" `(1/2 xx 4)`, which gives you the answer of 2. So "of" becomes "multiplied by" (or "times").

Using `=` for "is" and `xx` for "of", we continue translating:

`stackrel 3 overbrace"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 "`2/3` of `x`" on the other, because we're subtracting 3 from "two-thirds of `x`", not just from two-thirds. (Remember: `+" and "-` separate terms, while `xx" and "-:` create them.)

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

`stackrel (2//3) overbrace"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!

`2/3xx x-3=3",        "` or `"        "2/3 x-3=3`.

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

`2/3 x - cancel(3)+cancel color(red)(3)=3+color(red)(3)`

`color(white)(cancel 3+ cancel 3-)2/3 x= 6`

then multiplying both sides by the reciprocal of `2/3`:

`cancel color(red)(3/2) xx cancel(2/3) x = color(red)(3/2) xx 6`

`color(white)(cancel (3/2) xx cancel(2/3)) x = 3/2 xx 6 = (3 xx 6)/2 = 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?

`color(white)=`"`"3 less than "2/3" of 9"`"

`=`"`3" less than 6"`"

`=`"`3`",

which is what we were hoping for.