Proportions and Percentages - GED Math

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Question

What is 25% of 240?

Answer

To find a percentage of a whole number, we will multiply the two together. So, we get

$25% \cdot 240$

$\frac{25}{100} \cdot 240$

$\frac{25}{100} \cdot \frac{240}{1}$

Now, the zero in 100 and the zero in 240 can cancel. We get

$\frac{25}{10} \cdot \frac{24}{1}$

$\frac{25 \cdot 24}{10 \cdot 1}$

$\frac{600}{10}$

$\frac{60}{1}$

Therefore, 25% of 240 is 60.

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Question

45% of what number is 9?

Answer

The rate is 45% and the comparative number is 9; the unknown is the original number that 9 is 45% of. The statement is "(nine) is (forty-five percent) of (some number)", so the variable stands for a number.

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Question

The ratio of students to instructors at a prestigious medical school is 25:6. The school employs 90 instructors, and there will be 240 students returning. How many new students must it limit itself to admitting in order to continue to maintain this student-instructor ratio?

Answer

Let be the maximum number of students admitted overall.

For the school to maintain a 25:6 student-teacher ratio with 90 instructors, the ratios $\frac{N}{90}$ and $\frac{25}{6}$ must be equal. Therefore, we solve the proportion statement for :

$\frac{N}{90} = \frac{25}{6}$

$\frac{N}{90} \cdot 90= \frac{25}{6} \cdot 90$

$N= \frac{2,250}{6} = 375$

Since 240 spots are already filled by returning students, the medical school can admit up to new students.

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Question

Thomas’s current yearly salary is $44,000 per year. After an 8% raise, what will his new salary be?

Answer

8% of Thomas's salary of $44,000 is

$$ 44,000 \times 0.08 = $3,520$ .

Add this to the original salary:

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Question

Solve:

$14%\ of\ 52=$

Answer

$\small 14%\ of \ 52=0.14\times 52=7.28$

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Question

At a party, there are 7 women with blonde hair, 9 women with brown hair, and 5 women with red hair. What is the proportion of women with blonde hair?

Answer

A proprotion is the number of a specific event (blonde women) divided by the total number of events (total number of women).

$\small \frac{7}{7+9+5}=\frac{7}{21}=\frac{1}{3}$

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Question

Solve:

$\small 23%\ of\ 73=$

Answer

$\small 23%\ of\ 73=0.23\times73=16.8$

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Question

A bag contains $\small 7$ green marbles, $\small 2$ red marbles, and $\small 5$ blue marbles. What is the proportion of red marbles?

Answer

The proportion is equal to the number of specific events (red marbles) divided by the total number of events (sum of all colors).

$\small P=\frac{2}{7+2+5}=\frac{2}{14}=\frac{1}{7}$

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Question

What is of ?

Answer

$17%\ of\ 29=0.17\times 29=4.9$

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Question

What is 40% of 80% of ?

Answer

80% of is $\frac{80}{100 } = \frac{80 \div 20 }{100 \div 20 } = \frac{4}{5}$ multiplied by , or $\frac{4}{5}x$ .

40% of this is $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ multiplied by $\frac{4}{5}x$ ; this is

$\frac{2}{5} \cdot \frac{4}{5}x = \frac{2 \cdot 4}{5 \cdot 5} \cdot x = \frac{8}{25}x$ .

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Question

Seven tenths of three fourths of is what percent of ?

Answer

Seven tenths of three fourths of is equal to

$\frac{7}{10} \cdot \frac{3}{4} \cdot x = \frac{21}{40}x$ .

$\frac{21}{40}$ of a number, converted to percent, is

$\frac{21}{40} \times 100 % = \frac{2,100}{40} % = 52.5 %$ .

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Question

Ten percent of a number is six more than 5 percent of this number. Find the number.

Answer

The key word "is" indicates equality.

Ten percent of a number:

Six more than 5 percent of this number:

Solve and simplify:

$x=\frac{6}{0.05}= 120$

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Question

Bob's Bargain Basement has computer tables on sale at 50% off the retail price. For the holiday, an additional 15% is deducted. If the computer table is $240, how much is the table on sale for during the holiday?

Answer

In order to calculate the amount of the purchase, you first must find the cost of the table at 50% off. This can be done by taking $\frac{1}{2}$ of the original cost.

$\frac{240}{2} = 120.00$

This could also be done by converting the percent to a decimal and then multiplying that decimal by the original cost.

Then, the next step would be to take 15% of the $120. This is done by converting the percent to a decimal and then multiplying by $120.

15% = .15 as a decimal

This additional savings of $18.00 would then be subtracted from the $120.00.

This is the amount that will be paid after all discounts and savings.

Note: Do not add the initial savings of 50% to the additional savings of 15% to get a total savings of 65% and then work your problem from there. That is not the correct way to work this problem!

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Question

A building that is casts a shadow that is long. At the same time, another tree casts a shadow that is long. How tall is that tree?

Answer

To solve this you would set up and solve a direct proportion:

$\frac{180}{20} = \frac{x}{3}$

$\frac{20x}{20} = \frac{540}{20}$

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Question

Currently, Brian pays $865 a month for his rent. Monthly rent is due to increase by 12% starting next month. Rounded to the nearest dollar, how much will he pay then?

Answer

12% of $865 is

$$865 \times 0.12 = $103.80$ .

Add that amount to the current rent:

This rounds to $969, the correct choice.

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Question

Sherry deposits $3,000 at 6% simple interest per year. How much will she have in this account after three years, assuming she does not make any withdrawals or deposits?

Answer

Use the simple interest formula

with :

$I = $ 3,000 \cdot 0.06 \cdot 3 = $540$

This is the simple interest earned. Add it to the original principal:

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Question

Greg deposits $9,000 at 7% simple interest per year. How much will he have in this account after eighteen months, assuming he does not make any withdrawals or deposits?

Answer

Use the simple interest formula

with (18 months = 1.5 years):

$I = $ 9,000 \cdot 0.07 \cdot 1.5 = $945$

This is the simple interest earned. Add it to the original principal:

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Question

The ratio of students to instructors at a prestigious law school is 15:4. If the school employs 120 instructors, then how many students total must it limit itself to enrolling in order to maintain this student-instructor ratio?

Answer

Let be the maximum number of students admitted.

For the school to maintain a 15:4 student-teacher ratio with 120 instructors, the ratios $\frac{N}{120}$ and $\frac{15}{4}$ must be equal. Therefore, we solve the proportion statement for :

$\frac{N}{120} = \frac{15}{4}$

$\frac{N}{120} \cdot 120= \frac{15}{4} \cdot 120$

$N = \frac{1,800}{4} = 450$

The maximum number of students it can admit is 450.

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Question

Which of the following statements does not follow from the statement $\frac{X}{5} = \frac{Y }{7}$ ?

Answer

From $\frac{X}{5} = \frac{Y }{7}$ we can conclude the following:

is true, because the cross-products of a proportion statement are true.

$\frac{5}{X} = \frac{7}{Y }$ is true, because if two ratios are equivalent, their reciprocals are equivalent.

$\frac{X+5}{5} = \frac{Y+7 }{7}$ can be demonstrated to be true as follows:

$\frac{X}{5} = \frac{Y }{7}$

$\frac{X}{5} + 1= \frac{Y }{7} + 1$

$\frac{X}{5} +\frac{5}{5} = \frac{Y }{7} + \frac{7}{7}$

$\frac{X+5}{5} = \frac{Y+7 }{7}$

The fourth statement is false:

$\frac{X}{5} = \frac{Y }{7}$

$\frac{X}{5} \cdot \frac{5}{Y} = \frac{Y }{7}\cdot \frac{5}{Y}$

$\frac{X }{Y} = \frac{5} {7}$ , which contradicts the statement that $\frac{7}{5} = \frac{X }{Y}$ .

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Question

Which of the following statements follows from the statement ?

Answer

Two ratios are equivalent if and only if their cross-products are equal. We look for the proportion statement whose cross-products are and . Of the four choices, only $\frac{8}{B} = \frac{9}{A}$ fits this criterion, so it is the correct choice.

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