Remember that for percentages, the key to setting up the problem is intelligent translation. The word "is" becomes $\displaystyle =$, "of" signals multiplication, "what" (and equivalent words) signal a variable ($\displaystyle x$).

Therefore, we can translate:

$\displaystyle 72$ is $\displaystyle 41$% of what number?

As...

$\displaystyle 72=0.41 * x$

To solve, divide both sides by $\displaystyle 0.41$:

$\displaystyle x=175.60975609756098$

Rounded, this is:

$\displaystyle 175.610$

$\displaystyle 16 \frac{2}{3} \%$ of a number is equal to

$\displaystyle 16 \frac{2}{3} \div 100 = \frac{50}{3} \div 100 = \frac{50}{3} \times \frac{1}{100} = \frac{50}{300} = \frac{1}{6}$ 

of the number.

The question becomes equivalent to asking this - 500 is $\displaystyle \frac{1}{6}$ of what number? 

We can find out by dividing 500 by $\displaystyle \frac{1}{6}$:

$\displaystyle 500 \div \frac{1}{6} = 500 \times 6 = 3,000$.

Jack improves by 80%, so we multiply his original score by 1.8 (the 1 to represent his earlier score and the .8 to add on his improvement). 40% times 1.8 equals 72%.

Jill's second test is 130% of her first test (that is, a 30% improvement). To find her new score we multiply her first score by 1.3. 50% times 1.3 equals 65%.

Thus, Jack's second test score is higher.

No calculation is necessary. The whole in (b) is less, so 1 is a greater portion of that whole than the whole in (a). This makes (b) greater.

$\displaystyle 0.2 \% \textrm{ of }2,000$ can be rewritten as $\displaystyle 0.002 \times 2,000$

$\displaystyle 0.002 \times 2,000= 4 > \frac{2}{5}$

No calculation is necessary. In (a), the part is greater than the whole, so the percent $\displaystyle M$ must be greater than 100. In (b) The part is less than the whole, so $\displaystyle N$ must be less than 100. Therefore, $\displaystyle M > 100 > N$

Percentage involves part over whole. The total number of fruit was 48. The number of bananas was 12. Therefore, you can make a fraction from those numbers: $\displaystyle \tfrac{12}{48}$. Then, to find the percentage, divide 12 by 48. This gives you 0.25. To find the percentage, move the decimal point to the right two places. This gives you 25%.

70% of 4,000 is equal to $\displaystyle 4,000 \times 0.70 = 2,800$, which is 

$\displaystyle \frac{2,800}{6,400} \times 100$ percent of 6,400.

Evaluate:

$\displaystyle \frac{2,800}{6,400} \times 100 = 0.4375 \times 100 = 43.75 = 43 \frac{3}{4}$

The correct response is $\displaystyle 43 \frac{3}{4} \%$

40% of a number is the number multplied by 0.40; 70% of the number is the number multiplied by 0.70.

40% of $\displaystyle N$ is $\displaystyle 0.40 \cdot N$; 70% of that is $\displaystyle 0.70 \cdot 0.40 \cdot N= 0.28N$

70% of $\displaystyle N$ is $\displaystyle 0.70 \cdot N$; 40% of that is $\displaystyle 0.40 \cdot 0.70 \cdot N = 0.28N$

Regardless of the value of $\displaystyle N$, the quantities are equal.

The greater of the two can be shown to depend on the value of $\displaystyle x$.

Example 1: $\displaystyle x = 20$

Then 25% of $\displaystyle x+100$ is equal to 

$\displaystyle 0.25 (x+100) = 0.25 (20+100) = 0.25 \cdot 120 = 30$

and 50% of $\displaystyle x$ is equal to 

$\displaystyle 0.5x = 0.5 \cdot 20 = 10$

This makes (A) greater.

Example 2: $\displaystyle x = 300$

Then 25% of $\displaystyle x+100$ is equal to 

$\displaystyle 0.25 (x+100) = 0.25 (300+100) = 0.25 \cdot 400 = 100$

and 50% of $\displaystyle x$ is equal to 

$\displaystyle 0.5x = 0.5 \cdot 300 = 150$

This makes (B) greater.

Therefore, insufficient information is given in the problem to determine which is the greater.