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

Question

Eva Abramson · Accepted Answer

See explanation. (Warning: detailed answer ahead!)

To convert the sentence piece by piece into a mathematical equation, let's put it in writing: #"Three less than two-thirds of a number is three."# The first thing to note is that the numbers are directly translateable: #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 concludes the values (known and unknown). The operations and symbols need to be translated next. 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.) Once "less than" is changed to "minus" and the related terms are flipped, we obtain #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 equation translated! #2/3xx x-3=3", "# or #" "2/3 x-3=3#. Next, add three to both sides to find the solution: #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# After all of that, we have determined our number, which is 9. How much is three less than two-thirds of nine? Let's double check. #color(white)=#"#"3 less than "2/3" of 9"#" #=#"#3" less than 6"#" #=#"#3#", which is precisely what we wanted.

Avery Alexander · Answer

The mathematical expression is #2/3n - 3 = 3#
The number is #9# 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# Add three to each and every number in the equation. #2n -9 = 9# #2n = 18# #n = 9# In the new mathematical expression, verify the solution: #2/3n - 3 = 3# #2/3(9) - 3 = 3# # 6 = 3 + 3# # 6 = 6#

Isaiah Anthony · Answer

undefined Let $ x $ be the number.

Translate the given statement into a mathematical expression:

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

To solve for $ x $, first, add 3 to both sides:

$$ \frac{2}{3}x = 6 $$

Then, multiply both sides by $ \frac{3}{2} $:

$$ x = 9 $$

So, the number is 9.