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.

40% of \(\displaystyle N\) is equal to \(\displaystyle 0.4N\); 60% of that is \(\displaystyle 0.6 \cdot 0.4N = 0.24N\)

50% of \(\displaystyle N\) is equal to \(\displaystyle 0.5N\); 50% of that is \(\displaystyle 0.5 \cdot 0.5N = 0.25N\)

Since \(\displaystyle N\) is positive and \(\displaystyle 0.25 > 0.24\), 

\(\displaystyle 0.25N > 0.24N\),

and (B) is greater.

Convert the percentage to a decimal:

\(\displaystyle 0.5\%=\frac{0.5}{100}=\frac{5}{1000}=0.005\)

Multiply by 1,000,000 to get our answer.

\(\displaystyle 0.005 \times 1,000,000 = 5,000\)