GMAT Math : Percents

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #21 : Percents

A number is divided by four; the quotient is divided by three; that quotient is multiplied by five; and the decimal point in that product is moved to the left one place. What percent of the original number is that final result?

Possible Answers:

\(\displaystyle 16 \frac{2}{3} \%\)

\(\displaystyle 416 \frac{2}{3} \%\)

\(\displaystyle 24 \%\)

\(\displaystyle 240 \%\)

\(\displaystyle 4 \frac{1}{6} \%\)

Correct answer:

\(\displaystyle 4 \frac{1}{6} \%\)

Explanation:

The best way to solve this is to start with the original number being 100. The result is the percent.

The result of subsequent divisions by four and three, multiplication by five, and shifting the decimal point one place left - or division by ten - is 

\(\displaystyle 100 \div 4 \div 3 \cdot 5 \div 10\)

or, equivalently,

\(\displaystyle 100 \cdot \frac{1}{4} \cdot \frac{1}{3} \cdot 5 \cdot \frac{1}{10}\).

This can be calculated as 

\(\displaystyle \frac{100}{1} \cdot \frac{1}{4} \cdot \frac{1}{3} \cdot \frac{5}{1} \cdot \frac{1}{10} = \frac{25}{6} = 4 \frac{1}{6}\).

So for any number at which we start out, the final result is \(\displaystyle 4 \frac{1}{6} \%\) of the original number.

Example Question #21 : Calculating Percents

Untitled

The above is an annual income tax table for single persons in a given state.

Mr. McKenzie earned a salary of $2,270 per month until August 1, when he received a 10% raise. He also received a $2,000 bonus and $511 in interest.

Assuming that there was no additional income, calculate Mr. McKenzie's income tax to the nearest dollar.

Possible Answers:

\(\displaystyle \$247\)

\(\displaystyle \$137\)

None

\(\displaystyle \$150\)

\(\displaystyle \$260\)

Correct answer:

\(\displaystyle \$137\)

Explanation:

Mr. McKenzie earned $2,270 per month for seven months, earning a total of

\(\displaystyle \$2,270 \times 7 = \$15,890\)

His raise on August 1 was 10%, making his new salary

\(\displaystyle \$2,270 + \$2,270 \times 0.1 = \$2,270 + \$2 27 = \$2, 497\) per month.

He earned this over the remaining five months, earning

\(\displaystyle \$2, 497 \times 5 = \$12,485\)

Add these and the bonus and interest to get 

\(\displaystyle \$15,890 + \$12,485 + \$2,000 + \$511 = \$30,886\).

This places Mr. McKenzie in the $20-40,000 income bracket, so he will pay $50 plus 0.8% of earnings above $20,000. This will be

\(\displaystyle \$50 + 0.008 \times (\$30,886 - \$20,000)\)

\(\displaystyle =\$50 + 0.008 \times \$10,886\)

\(\displaystyle \approx \$50 + \$87.08\)

\(\displaystyle = \$137.08\)

The correct response is $137.

Example Question #21 : Calculating Percents

Untitled

The above is an annual income tax table for single persons in a given state.

Between January 1 and July 1, Mr. Smith earned $722 per month from a part time job. He earned a raise beginning on July 1, but he ended up not paying any income tax for the year. Assuming he had no other income, what percent was the highest possible raise in his salary? Choose the answer that is closest to the maximum possible raise.

Possible Answers:

\(\displaystyle 25 \%\)

\(\displaystyle 20 \%\)

\(\displaystyle 30 \%\)

\(\displaystyle 40 \%\)

\(\displaystyle 35 \%\)

Correct answer:

\(\displaystyle 30 \%\)

Explanation:

Since Mr. Smith did not pay any income tax, his income was at most $10,000. 

He earned $722 per month over six months for a total of 

\(\displaystyle \$722 \times 6 = \$4,332\)

The maximum he earned over the remaining six months is

\(\displaystyle \$10,000 - \$4,332 = \$5,668\), which is

\(\displaystyle \$5,668 \div 6 = \$944.67\) per month.

This represents a 

\(\displaystyle \frac{\$944.67 - \$722 }{\$722 } \times 100 \% = \frac{\$222.67 }{\$722 } \times 100 \% \approx 31 \%\)

maximum raise.

The closest choice is 30%.

Example Question #23 : Calculating Percents

Untitled

The above is an annual income tax table for married couples in a given state.

Mr. Barrett began a job on March 1 that paid a salary of \(\displaystyle \$3,872\) per month, and remained with it through the remainder of the year. Mrs. Barrett woked at a salary of \(\displaystyle \$2,882\) per month from January 1 until she was laid off on August 31; her layoff ended on December 1, and she resumed work at her former salary. The couple also received interest income of \(\displaystyle \$298\)

Assuming that there was no additional income, calculate the Barretts' income tax to the nearest dollar.

Possible Answers:

\(\displaystyle \$1,104\)

\(\displaystyle \$603\)

\(\displaystyle \$841\)

\(\displaystyle \$505\)

\(\displaystyle \$554\)

Correct answer:

\(\displaystyle \$554\)

Explanation:

Mr. Barrett earned $3,872 per month for ten months for a total of

\(\displaystyle \$3,872 \times 10 = \$38,720\)

Mrs. Barrett earned $2,882 per month for nine months for a total of

\(\displaystyle \$2,882 \times 9= \$25,938\)

These salaries and the interest add up to

\(\displaystyle \$38,720 +\$25,938+\$298 = \$64,956\)

This puts the Barretts in the $60-80,000 range, so their tax is $470 plis 1.7% of earnings above $60,000. This is

\(\displaystyle \$470 + 0.017 \times \left (\$64,956 - \$60,000 \right )\)

\(\displaystyle =\$470 + 0.017 \times \$4,956\)

\(\displaystyle =\$470 + 0.017 \times \$4,956\)

\(\displaystyle =\$470 + \$84.25\)

\(\displaystyle =\$554.25\)

The correct response is $554.

Example Question #24 : Calculating Percents

Untitled

The above is an annual income tax table for single persons in a given state.

Mr. Parsons earned a salary of \(\displaystyle \$3,122\) per month for the entire year. He also earned \(\displaystyle \$886\) per month from April 1 to November 30 from a part-time job, and \(\displaystyle \$817\) from interest. Assuming Mr. Parsons had no other income, calculate Mr. Parsons's income tax to the nearest dollar.

Possible Answers:

\(\displaystyle \$443\)

\(\displaystyle \$280\)

\(\displaystyle \$326\)

\(\displaystyle \$153\)

\(\displaystyle \$198\)

Correct answer:

\(\displaystyle \$280\)

Explanation:

Mr. Parsons earned $3,122 per month for twelve months in his main job for a total of 

\(\displaystyle \$3,122 \times 12 = \$37,464\)

He earned $886 per month for eight months in a part-time job for a total

\(\displaystyle \$886 \times 8 = \$7,088\)

Add these, and the $817 he earned from interest:

\(\displaystyle \$37,464 + \$7,088 + \$817= \$45,369\)

This puts Mr. Parsons in the $40-60,000 tax bracket, so he will pay $210 plus 1.3% of what he earns above $40,000. This will be

\(\displaystyle \$ 210+ 0.013 \times (\$ 45,369 - \$ 40,000)\)

\(\displaystyle = \$ 210+ 0.013 \times \$ 5,369\)

\(\displaystyle \approx \$ 210+ \$ 69.80\)

\(\displaystyle \approx \$ 279.80\)

This rounds to $280.

Example Question #21 : Percents

Total sales at XYZ Corporation increased by \(\displaystyle 25\%\) this year. What is the amount of sales for this year if last year's sales were \(\displaystyle \$200,000\)?

Possible Answers:

\(\displaystyle \$250,000\)

\(\displaystyle \$125,000\)

\(\displaystyle \$450,000\)

\(\displaystyle \$225,000\)

\(\displaystyle \$50,000\)

Correct answer:

\(\displaystyle \$250,000\)

Explanation:

The amount of sales for this year is 125% of last year's sales. Therefore, we can calculate in the following way:

\(\displaystyle 200000\times1.25=250,000\)

Or:

\(\displaystyle 200,000+0.25\times200000=200,000+\frac{200000}{4}=250000\)

Example Question #105 : Arithmetic

\(\displaystyle M\) is 35% of \(\displaystyle N\)\(\displaystyle L\) is 25% of \(\displaystyle N\). What percent of \(\displaystyle M\) is \(\displaystyle L\)?

Possible Answers:

\(\displaystyle 140 \%\)

\(\displaystyle 28\frac{4}{7} \%\)

\(\displaystyle 71\frac{3}{7} \%\)

\(\displaystyle 40 \%\)

Insufficient information is given to answer the question.

Correct answer:

\(\displaystyle 71\frac{3}{7} \%\)

Explanation:

\(\displaystyle M\) is 35% of \(\displaystyle N\),and \(\displaystyle L\) is 25% of \(\displaystyle N\), so

\(\displaystyle M = 0.35 N\) and \(\displaystyle L = 0.25N\).

To find out what percent \(\displaystyle L\) is of \(\displaystyle M\), evaluate:

\(\displaystyle \frac{L}{M} \times 100 \% = \frac{0.25N}{0.35N} \times 100 \% = \frac{ 25 }{ 35 } \times 100 \% = \frac{ 2,500 }{ 35 } \% = 71\frac{3}{7} \%\)

Example Question #101 : Arithmetic

\(\displaystyle M\) is 40% of \(\displaystyle N\)

\(\displaystyle \frac{2}{M}\) is what percent of \(\displaystyle \frac{3}{N}\) ?

Possible Answers:

\(\displaystyle 26 \frac{2}{3} \%\)

\(\displaystyle 60 \%\)

Insufficient information is given to answer the question.

\(\displaystyle 375 \%\)

\(\displaystyle 166 \frac{2}{3} \%\)

Correct answer:

\(\displaystyle 166 \frac{2}{3} \%\)

Explanation:

\(\displaystyle M\) is 40% of \(\displaystyle N\), so \(\displaystyle M = 0.40 N = \frac{2}{5}N = \frac{2N}{5}\).

\(\displaystyle \frac{2}{M} = \frac{2}{\frac{2N}{5}} = 2 \div \frac{2N}{5} = 2 \cdot \frac{5}{2N}= \frac{5}{N}\)

The question can therefore be rewritten as follows:

\(\displaystyle \frac{5}{N}\) is what percent of \(\displaystyle \frac{3}{N}\) ?

Solve by evaluating:

\(\displaystyle \frac{\frac{5}{N}}{\frac{3}{N}} \times 100 \%\)

\(\displaystyle =\left (\frac{5}{N} \div \frac{3}{N} \right )\times 100 \%\)

\(\displaystyle =\left (\frac{5}{N} \cdot \frac{N}{3} \right )\times 100 \%\)

\(\displaystyle = \frac{5} {3} \times 100 \%\)

\(\displaystyle = 166 \frac{2}{3} \%\)

Example Question #21 : Percents

\(\displaystyle 6M\) is 35% of \(\displaystyle 4N\). What percent of \(\displaystyle M\) is \(\displaystyle N\)?

Possible Answers:

\(\displaystyle 52 \frac{1}{2} \%\)

\(\displaystyle 428\frac{4}{7} \%\)

\(\displaystyle 840 \%\)

\(\displaystyle 11\frac{38}{42} \%\)

\(\displaystyle 23\frac{1}{3} \%\)

Correct answer:

\(\displaystyle 428\frac{4}{7} \%\)

Explanation:

\(\displaystyle 6M\) is 35% of \(\displaystyle 4N\), so 

\(\displaystyle 6M = \frac{35}{100} \cdot 4N = \frac{7}{5}N\)

\(\displaystyle \frac{7}{5}N = 6M\), so

\(\displaystyle \frac{5}{7} \cdot \frac{7}{5}N =\frac{5}{7} \cdot 6M\)

\(\displaystyle N =\frac{30}{7} M\)

So \(\displaystyle N\) is \(\displaystyle \frac{30}{7}\), or

\(\displaystyle \frac{30}{7} \times 100 \% = 428\frac{4}{7} \%\) 

of \(\displaystyle M\).

Example Question #108 : Arithmetic

\(\displaystyle X\) is 40% of \(\displaystyle N\), and \(\displaystyle N\) is 75% of \(\displaystyle Y\)\(\displaystyle N\) and \(\displaystyle Y\) are positive integers.

True or false: \(\displaystyle X\) is a positive integer.

Statement 1: \(\displaystyle N\) is a multiple of 5.

Statement 2: \(\displaystyle Y\)is a multiple of 4.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

Assume Statement 1 alone. \(\displaystyle X\) is 40% of \(\displaystyle N\), or, equivalently, \(\displaystyle X = \frac{2}{5}N\)\(\displaystyle N\) is a multiple of 5, so for some integer \(\displaystyle M\)\(\displaystyle N = 5M\). Consequently, 

\(\displaystyle X = \frac{2}{5} \cdot 5M = 2M\).

\(\displaystyle X\) is twice an integer so \(\displaystyle X\) is itself an integer.

Statement 2 alone is inconclusive.

For example, if \(\displaystyle Y = 8\),  \(\displaystyle N\) is 75% of this, which is 

\(\displaystyle N = 0.75 \cdot 8 = 6\),

and \(\displaystyle X\) is 40% of 6, which is 

\(\displaystyle X = 0.40 \cdot 6 =2.4\)

which is not an integer.

But if \(\displaystyle Y = 20\),  \(\displaystyle N\) is 75% of this, which is 

\(\displaystyle N = 0.75 \cdot 20 =15\),

and \(\displaystyle X\) is 40% of 15, which is 

\(\displaystyle X = 0.40 \cdot 15 = 6\)

which is an integer.

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