Arithmetic - GMAT Quantitative

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Question

The Duchy of Grand Fenwick uses an unusual currency system. It takes 24 tiny fenwicks to make a big fenwick.

At current exchange rates, 1 big fenwick can be exchanged for $3.26 American currency. For how much in Grand Fenwick currency can an American tourist exchange $300 (rounded to the nearest tiny fenwick)?

Answer

$3.26 can be exchanged for 1 big fenwick, or, equivalently, 24 tiny fenwicks. We can set up a proportion statement, where is the number of tiny fenwicks for which $300 can be exchanged:

$\frac{F}{300} = \frac{24}{3.26}$

Solve for :

$\frac{F}{300} \cdot 300 = \frac{24}{3.26} \cdot 300$

$F= \frac{24\cdot 300}{3.26} \approx 2,209$

The answer is 2,209 tiny fenwicks, which can be counted up with division:

$2,209 \div 24 = 92 \textrm{ R 1}$

or 92 big fenwicks and 1 tiny fenwick.

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Question

Five different pizza places offer five different specials. Assuming that all of these pizzas are of the same thickness and that all are of the same quality, which of the following is the best buy?

Answer

Since all of the pizzas are of the same thickness and quality, to determine the best bargain, calculate the price per square inch of each. The least amount will mark the best bargain.

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A round pizza 10 inches in diameter for $5.99:

The area of the pizza in square inches is $A = \pi r^{2} = \pi \cdot5^{2} = 25\pi$

The cost per square inch: $5.99\div25\pi =0.076$ or 7.6 cents

A round pizza 12 inches in diameter for $8.99

The area of the pizza in square inches is $A = \pi r^{2} = \pi \cdot6^{2} =36\pi$

The cost per square inch: $8.99\div36\pi =0.079$ or 7.9 cents

A 9 inch by 9 inch square pizza for $6.99

The area of the pizza in square inches is $9 \cdot 9 = 81$

The cost per square inch: $6.99 \div 81 = 0.086$ or 8.6 cents

A 10 inch by 10 inch square pizza for $7.99

The area of the pizza in square inches is $10 \cdot 10 = 100$

The cost per square inch: $7.99 \div 100 = 0.080$ or 8.0 cents

A 12 inch by 8 inch rectangular pizza for $9.99:

The area of the pizza in square inches is $12 \cdot 8 = 96$

The cost per square inch: $9.99 \div 96 = 0.104$ or 10.4 cents

The round pizza 10 inches in diameter for $5.99 is the best buy.

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Question

127 people signed up for a subscription service. 45 signed up for weekly deliveries, 34 for monthly deliveries, and the rest for annual deliveries. What is the ratio of weekly subscribers to annual subscribers?

Answer

Let's first find the number of annual subscribers:

The ratio of weekly subscribers to annual subscribers is therefore .

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Question

A furlong is a measure of length used in horse racing. Eight furlongs are equal to one mile.

Express 20 furlongs in feet.

Answer

Divide 20 furlongs by 8 furlongs per mile, then multiply by 5,280 feet per mile.

$\frac{20}{8} \times 5,280 = 2.5 \times 5,280 = 13,200$

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Question

A fathom is a unit of underwater depth equal to six feet.

Express two miles in fathoms.

Answer

Multiply 2 miles by 5,280 feet per mile, then divide by 6 feet per fathom:

$(2 \times 5,280) \div 6 = 10,560 \div 6 = 1,760 \textrm{ fathoms}$

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Question

A clothing store is having a sales for the holiday season. They mark down all items by $43\frac{4}{5}%$ . What is the sales price of a suit whose original price is $\$340$ ?

Answer

We start by finding the value of the mark down:

$43\frac{4}{5}=43+\frac{4}{5}=43+0.80=43.8%$

Therefore, the mark down is 43.8% or 0.438, which means that the new price of the suit is:

$340\cdot (1-0.438)=191.08$

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Question

Noah and Lilly both work for Company ABC. Lilly earns more than Noah, and each year every employee gets a salary increase. In 2012, Noah's salary was $\$60,000$ .

What was Lilly's salary in 2013?

Answer

We start by finding Lilly's salary in 2012:

$60,000 x 1.20 = $60,000 + $12,000 = $72,000

Then we account for the annual 10% increase to find Lilly's salary in 2013:

$72,000 x 1.10 = $72,000 + $7,200 = $79,200

So Lilly's salary in 2013 was $79,200.

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Question

What is the sum of the prime numbers that are greater than 50 but less than 60?

Answer

A prime number is only divisible by the number 1 and itself. Of the integers between 50 and 60, all of the even integers are also divisible by the number 2 so they are not prime numbers. The integers 51 and 57 are divisible by 3. The integer 55 is divisible by 5. The only integers that are prime are 53 and 59. The sum of these two integers is 112.

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Question

What is the product of the four smallest prime numbers?

Answer

We must remember that 0 and 1 are NOT prime numbers, but 2 is.

The four smallest prime numbers are 2, 3, 5, and 7. Then 2 * 3 * 5 * 7 = 210.

Note: There are NO negative prime numbers, so we don't have to look for tiny, negative numbers here.

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Question

If a and b are even integers, what must be odd?

Answer

The sum (or difference) or 2 even integers is even. Similarly, the product (or quotient) of 2 even integers is also even; therefore the answer must be $\dpi{100} \small a+b-1$ , which can be easily checked by plugging in any two even numbers.

For example, if $\dpi{100} \small a=2\ and\ b=4,\ a+b-1=2+4-1=5$ , which is odd.

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Question

Three consecutive numbers add up to 36. What is the smallest number?

Answer

The sum of 3 consecutive numbers would be

which simplifies into

Set that equation equal to 36 and solve.

$3n + 3 = 36 \rightarrow 3n = 33 \rightarrow n = 11$

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Question

Becky has to choose from 4 pairs of pants, 6 shirts, and 2 pairs of shoes for an interview. If an outfit consists of 1 pair of pants, 1 pair of shoes, and 1 shirt, how many options does she have?

Answer

To find total number, multiply the number of each item.

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Question

What is the greatest prime factor of

Answer

The only prime factors are 3 and 5, therefore, 5 will be the greatest prime factor.

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Question

Solve:

$(8\times 10^6)+(5\times 10^4)+(3\times 10^3)+(4\times 10^2)+(6\times 10)+(7\times 10^0)=$

Answer

$8\times 10^6=8,000,000$

$5\times 10^4=50,000$

$3\times 10^3=3,000$

$4\times 10^2=400$

$6\times 10=60$

$7\times 10^0=7$

The sum is 8,053,467

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Question

For how many integers, , is $\frac{x-1}{x-2}$ an integer?

Answer

Since the numerator is always 1 greater than the denominator, we know that for large enough values of , it's never going to be an integer (one will be even, other odd). In fact, there are only 2 cases where this can be done. The first is dividing 0. Since 0 is divisible by every number, if the numerator is 0, then we will still get an integer. Thus one answer is $x=1 \Rightarrow \frac{(1)-1}{(1)-2}=\frac{0}{-1} = 0$

The other answer occurs as a special case as well. We can divide any number by 1 evenly, so when the denominator is 1 we get an integer:

$x=3 \Rightarrow \frac{(3)-1}{(3)-2} = \frac{2}{1} = 2$

In every other case, we will have a non-integer.

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Question

A positive integer divided by 9 yields remainder 7. What remainder does $N^{2}$ divided by 9 yield?

Answer

The easiest way to explain this is by example.

We can use 16, since $16 \div 9 = 1 \textrm{ R } 7$

$16 ^{2 } = 256$

$256 \div 9 = 28 \textrm{ R } 4$

This makes 4 the correct choice.

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Question

If a positive integer is divided by another positive integer , then the quotient is 6, and there is no remainder.

Which of these choices is a possible value of ?

Answer

The conditions of the problem can be rewritten as $N \div M = 6$ , or .

, meaning that the sum of the two numbers is a multiple of 7. Each of the given choices is a multiple of 7, so any of them can be .

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Question

A positive integer divided by 5 yields remainder 3. What remainder does $N^{3}$ divided by 5 yield?

Answer

Any integer that is divisible by 5 must end in a 0 or a 5, so any integer which, when divided by 5, yields remainder 3 must end in 3 or 8. Since the cubes of 3 and 8 are 27 and 512, the cube of an integer that ends in one of these digits must end in 2 or 7, meaning that, when divided by 5, the remainder will be 2.

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Question

What is the first digit in the base-six representation of the number 936?

Answer

, or, equivalently, $6^{3}< 936 < 6^{4}$ . This makes 936 a four-digit number when written in base six. The first digit is equal to the number of times 216 divides into 936.

$936 \div 216 = 4 \textrm{ R } 72$ ,

4 is the first digit of this number.

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Question

What is the last digit in the base-eight representation of the number 735?

Answer

Divide 735 by 8. The remainder is the last digit of the base-eight representation.

$735 \div 8 = 91 \textrm{ R } 7$

The correct choice is 7.

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