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

##### 2 Answers

The mathematical expression is

The number is

#### Explanation:

Let the unknown number be shown as

Three less than

All of this is equal to

#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#

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

#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 "

Similarly, the word "of" becomes multiplication. For example, if I asked you, *times* four"

Using

#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 (

Think about it: what's one less than seven? Six, right? But you didn't find that by subtracting *swap the order of the two terms*, and then put a minus sign between them. In other words, "

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 "*(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

#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.