AdditionSubtractionMultiplicationDivisionEqualsPowerRootMultiplied by 2Divided by 2:InequalityUnknown quantity+−×,⋅/,÷,ba,%=a2,a3,a4a,3n×2÷2,2a>≥<≤xplus, added to, more than, greater than,sum, total, increased byminus, subtracted from, less than, fewerdifference, decreased by, reduced bytimes, of, product, multiplied bydivided by, quotient, per, the ratio ofis, will be, is equal to, is the same as,the result of, yieldsthe square of, squared, the cube of, cubedraised to the fourth powersquare-rooted, cube-rooted, the third roottwice, doubled, two times,twice as much ashalf of, half as much as, halvedone-half timesmore than, greater thanat leastfewer, less thanat mostwhat, how many, how much, a number
(A) Three
(B) Four
(C) Seven
(D) Eleven
(E) Twenty eight
Correct Answer: A
Solution:
We replace the line with the variable x.
Translating the words into math, we get :
4Four=is7−x x less than 7
We can now create an equation:
4x+4xx====7−x77−43translate the words into mathadd x to both sidessubtract 4 from both sidessolve
Incorrect Choices:
(B) Tip: Just because a number appears in the question doesn’t mean it is the answer.
(C) Tip: Just because a number appears in the question doesn’t mean it is the answer.
(D) Tip: Read the entire question carefully.
You will get this answer if you translate the phrase as
4Four=isxx−less than7seven
(E)
This answer is offered to confuse you. It is the product of 4 and 7.
A
B
C
D
E
Which of the following is equal to half of 37 percent of 540?
(A) 21⋅37⋅540
(B) 37%of540
(C) 3721%of540
(D) 37%of270
(E) 21⋅37%of270
How much greater than m−5 is m+7?
(A) 2
(B) 5
(C) 7
(D) 12
(E) 35
Correct Answer: D
Solution 1:
Let m+7 be x greater than m−5. First we translate the words into math:
xHow much+greater thanm−5=ism+7?
We can now create an equation:
x+m−5x−5xx====m+777+512translate the words into mathsubtractmfrom both sidesadd5to both sides7+5=12(1)(2)(3)(4)
Solution 2:
Tip: Look for short-cuts. m+7 is greater than m−5. To find by how much, we subtract m−5 from m+7:
(m+7)−(m−5)=m+7−m+5=12
Solution 3:
Tip: Look for short-cuts. Tip: Replace variables with numbers.
Let m=0. Then the question becomes: How much greater than 0−5 is 0+7, or, how much greater than −5 is 7? The answer is 7−(−5)=12.
Solution 4:
We can use the number line to solve the problem. Select any point for m. The point 7 tick marks to the right of m will be m+7. The point 5 tick marks to the left of m will be m−5. The number of segments between m−5 and m+7 is 12, and therefore, m+7 is 12 greater than m−5.
Incorrect Choices:
(A) Tip: Be careful with signs!
You might get this wrong answer if in step (3) of Solution 1 you add −5 to both sides, not 5, as shown:
−5+x=7x=7-5mistake: added−5to both sides(2)(3)
Similarly, in Solution 2 and Solution 3, you could make an error when distributing the negative sign:
The difference of 5n and (2m)22msquared=is equal to the square root of the sum of n2nsquared and m3mcubed.
5n−(2m)2The difference of5nand(2m)25n−(2m)25n−(2m)2⇓=⇓=⇓=the square root ofn2+m3the sum ofn2andm3n2+m3the square root ofn2+m3n2+m3
Incorrect Choices:
(A)
Squared means raised to the power of 2. Cubed means raised to the power of 3. The mistake here is that 2m is multiplied by 2 instead of raised to the power of 2, n is multiplied by 2 instead of raised to the power of 2, and m is multiplied by 3 instead of raised to the power of 3.
(B)
The right side of the equation ignores the instructions. n should be squared, and m should be cubed.
(C)
The mistake here is that the two terms, n2 and m3, are square-rooted individually, and then the sum of the square roots is found. However, the problem instructs that we find the sum n2+m3 first and then square-root it.
(E)
Here, the difference of 5n−2m is squared, but according to the problem, we should subtract 2m squared from 5n.
Review
If you thought these examples difficult and you need to review the material, these links will help: