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

2 Answers
Feb 9, 2017

Answer:

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#

Feb 9, 2017

Answer:

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.