Rational Number Operations - Accuplacer Qas

A rectangular garden has a length of 5.5 meters and a width of $3\frac{1}{4}$ meters. Fencing costs $8 per meter. What is the total cost to build a fence around the entire garden?
Answer: $140.00 First, calculate the perimeter of the garden. The formula for the perimeter of a rectangle is $P = 2(L+W)$ . Convert the width to a decimal: $3\frac{1}{4} = 3.25$ meters. Now find the perimeter: $P = 2(5.5 + 3.25) = 2(8.75) = 17.5$ meters. Finally, calculate the total cost by multiplying the perimeter by the cost per meter: $17.5 \times 8 = \$140.00$ .
A recipe for fruit punch calls for mixing cranberry juice and apple juice in the ratio $2:3$ . If the total amount of punch needed is $3\frac{3}{4}$ quarts, and cranberry juice costs $2.40 per quart while apple juice costs $1.80 per quart, what is the total cost of the juice needed?
Answer: $7.65 The ratio $2:3$ means cranberry juice makes up $\frac{2}{5}$ of the total and apple juice makes up $\frac{3}{5}$ . Total punch needed: $3\frac{3}{4} = \frac{15}{4}$ quarts. Cranberry juice needed: $\frac{2}{5} \times \frac{15}{4} = \frac{30}{20} = \frac{3}{2} = 1.5$ quarts. Apple juice needed: $\frac{3}{5} \times \frac{15}{4} = \frac{45}{20} = \frac{9}{4} = 2.25$ quarts. Cost calculation: Cranberry juice cost = $1.5 \times \$2.40 = \$3.60$ . Apple juice cost = $2.25 \times \$1.80 = \$4.05$ . Total cost = $\$3.60 + \$4.05 = \$7.65$ . Choice B results from incorrectly calculating the ratio as $\frac{2}{3}$ and $\frac{1}{3}$ instead of $\frac{2}{5}$ and $\frac{3}{5}$ . Choice C comes from arithmetic errors in the multiplication. Choice D results from using the wrong total amount or swapping the juice prices.
A carpenter has a piece of wood that is $12\frac{1}{2}$ feet long. She cuts off a piece that is $3\frac{3}{4}$ feet long. She then wants to cut the remaining piece into 5 equal-length shelves. What is the length of each shelf in feet?
Answer: $1\frac{3}{4}$ First, find the length of the remaining piece of wood by subtracting the cut piece from the original length: $12\frac{1}{2} - 3\frac{3}{4} = \frac{25}{2} - \frac{15}{4} = \frac{50}{4} - \frac{15}{4} = \frac{35}{4}$ feet. Next, divide the remaining length by the number of shelves, 5: $\frac{35}{4} \div 5 = \frac{35}{4} \times \frac{1}{5} = \frac{35}{20} = \frac{7}{4} = 1\frac{3}{4}$ feet.
A carpenter cuts a board that is $4\frac{3}{8}$ feet long into three pieces. The first piece is $1\frac{5}{6}$ feet long, and the second piece is $0.75$ feet long. What is the length of the third piece in feet?
Answer: $1\frac{19}{24}$ feet The third piece equals the total length minus the sum of the first two pieces. Convert all measurements to the same form: $4\frac{3}{8} = \frac{35}{8}$ , $1\frac{5}{6} = \frac{11}{6}$ , and $0.75 = \frac{3}{4}$ . Find the sum of the first two pieces using LCD 12: $\frac{11}{6} = \frac{22}{12}$ and $\frac{3}{4} = \frac{9}{12}$ , so $\frac{22}{12} + \frac{9}{12} = \frac{31}{12}$ . Now subtract from the total using LCD 24: $\frac{35}{8} = \frac{105}{24}$ and $\frac{31}{12} = \frac{62}{24}$ . Therefore, $\frac{105}{24} - \frac{62}{24} = \frac{43}{24} = 1\frac{19}{24}$ . Choice B results from converting 0.75 incorrectly to $\frac{5}{8}$ . Choice C comes from calculation errors when finding the LCD. Choice D results from mishandling the mixed number conversion.
A number $n$ is decreased by $\frac{1}{5}$ of its value. The result is then multiplied by $-2.5$ . If the final result is 12, what was the original number $n$ ?
Answer: -6 Work backwards from the final result. The result after multiplication is 12. Before being multiplied by -2.5, the number was $12 \div (-2.5) = -4.8$ . This number, -4.8, is the result of decreasing $n$ by $\frac{1}{5}$ of its value. This means $n - \frac{1}{5}n = -4.8$ , which simplifies to $\frac{4}{5}n = -4.8$ . To find $n$ , divide -4.8 by $\frac{4}{5}$ : $n = -4.8 \div \frac{4}{5} = -4.8 \times \frac{5}{4} = -1.2 \times 5 = -6$ .
The product of two numbers is $-0.75$ . If one of the numbers is $4\frac{1}{2}$ , what is the other number?
Answer: $-\frac{1}{6}$ Let the unknown number be $x$ . The equation is $4\frac{1}{2} \times x = -0.75$ . Convert both numbers to fractions to make the division easier. $4\frac{1}{2} = \frac{9}{2}$ and $-0.75 = -\frac{3}{4}$ . The equation is $\frac{9}{2}x = -\frac{3}{4}$ . To solve for $x$ , divide $-\frac{3}{4}$ by $\frac{9}{2}$ , which is the same as multiplying by its reciprocal: $x = -\frac{3}{4} \times \frac{2}{9} = -\frac{6}{36}$ . Simplifying this fraction gives $-\frac{1}{6}$ .
A recipe for cookies requires $1\frac{3}{4}$ cups of flour and makes 2 dozen cookies. A baker wants to make 5 dozen cookies. How many cups of flour are needed?
Answer: $4\frac{3}{8}$ First, find the scaling factor for the recipe by dividing the desired amount by the original amount: $5 \div 2 = 2.5$ or $\frac{5}{2}$ . Next, multiply the amount of flour by this scaling factor. Convert $1\frac{3}{4}$ to an improper fraction, which is $\frac{7}{4}$ . Now multiply: $\frac{7}{4} \times \frac{5}{2} = \frac{35}{8}$ . Converting back to a mixed number gives $4\frac{3}{8}$ cups.
Let $P = (-2) \div \frac{1}{4}$ and $Q = (-\frac{1}{2}) \times 4$ . Which of the following statements is true?
Answer: $Q > P$ First, calculate the value of P: $P = -2 \div \frac{1}{4} = -2 \times 4 = -8$ . Next, calculate the value of Q: $Q = -\frac{1}{2} \times 4 = -2$ . Now compare the two values. Since -2 is greater than -8, the true statement is $Q > P$ .
If $x = -\frac{1}{2}$ and $y = \frac{2}{3}$ , what is the value of the expression $4x - 6y$ ?
Answer: -6 Substitute the given values for $x$ and $y$ into the expression. First, calculate $4x$ : $4(-\frac{1}{2}) = -2$ . Next, calculate $6y$ : $6(\frac{2}{3}) = \frac{12}{3} = 4$ . Finally, substitute these values back into the expression $4x - 6y$ : $-2 - 4 = -6$ .
What is the product of the opposite of $-2.5$ and the reciprocal of $\frac{5}{4}$ ?
Answer: 2 First, identify the two numbers. The opposite of -2.5 is 2.5. The reciprocal of $\frac{5}{4}$ is $\frac{4}{5}$ . Next, find their product. It is helpful to convert 2.5 to a fraction: $2.5 = \frac{5}{2}$ . Now multiply: $\frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2$ .
The sum of three numbers is $4\frac{1}{8}$ . If two of the numbers are $-1.5$ and $\frac{5}{4}$ , what is the third number?
Answer: $4\frac{3}{8}$ Let the third number be $x$ . The equation is $-1.5 + \frac{5}{4} + x = 4\frac{1}{8}$ . Convert all numbers to fractions with a common denominator of 8. $-1.5 = -\frac{3}{2} = -\frac{12}{8}$ . $\frac{5}{4} = \frac{10}{8}$ . The sum is $4\frac{1}{8} = \frac{33}{8}$ . The equation becomes $-\frac{12}{8} + \frac{10}{8} + x = \frac{33}{8}$ , which simplifies to $-\frac{2}{8} + x = \frac{33}{8}$ . To find $x$ , add $\frac{2}{8}$ to both sides: $x = \frac{33}{8} + \frac{2}{8} = \frac{35}{8}$ . As a mixed number, this is $4\frac{3}{8}$ .
Sam had $240.00 in his bank account. He spent $\frac{1}{8}$ of the money on a concert ticket. He then deposited a check for $67.50. Finally, he withdrew $\frac{1}{3}$ of the new balance. How much money is left in his account?
Answer: $185.00 First, calculate the cost of the ticket: $\frac{1}{8} \times 240 = \$30$ . The new balance is $\$240 - \$30 = \$210$ . Next, add the deposit: $\$210 + \$67.50 = \$277.50$ . Then, calculate the withdrawal amount: $\frac{1}{3} \times 277.50 = \$92.50$ . Finally, find the remaining balance: $\$277.50 - \$92.50 = \$185.00$ .
The temperature at noon was $5.4^\circ$ C. By 6:00 PM, it had dropped by $8\frac{1}{2}$ degrees. By midnight, it had dropped another $2.3^\circ$ C. What was the temperature at midnight?
Answer: $-5.4^\circ$ C Start with the initial temperature and subtract the two drops. It's easiest to convert the fraction to a decimal: $8\frac{1}{2} = 8.5$ . The calculation is $5.4 - 8.5 - 2.3$ . First, $5.4 - 8.5 = -3.1$ . Then, $-3.1 - 2.3 = -5.4$ . The temperature at midnight was $-5.4^\circ$ C.
What is the value of $\frac{2}{3} \left( \frac{9}{4} - \frac{1}{2} \right)$ ?
Answer: $\frac{7}{6}$ Following the order of operations, first calculate the value inside the parentheses. Find a common denominator for the fractions: $\frac{9}{4} - \frac{1}{2} = \frac{9}{4} - \frac{2}{4} = \frac{7}{4}$ . Now, multiply this result by $\frac{2}{3}$ : $\frac{2}{3} \times \frac{7}{4} = \frac{14}{12}$ . Simplify the fraction by dividing the numerator and denominator by 2, which gives $\frac{7}{6}$ .
A tank is $\frac{3}{4}$ full of water. After 15.5 gallons are used, the tank is $\frac{1}{3}$ full. What is the total capacity of the tank in gallons?
Answer: 37.2 Let $C$ be the total capacity of the tank. The change in the amount of water is the fraction of the tank that was drained: $\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}$ . This fraction of the total capacity is equal to the 15.5 gallons that were used. So, $\frac{5}{12}C = 15.5$ . To find $C$ , multiply 15.5 by the reciprocal of $\frac{5}{12}$ : $C = 15.5 \times \frac{12}{5} = 3.1 \times 12 = 37.2$ gallons.
A stock's price started at $45. On Monday, it lost $\frac{1}{10}$ of its value. On Tuesday, the new price gained $\frac{1}{5}$ of its value. What was the price of the stock at the end of Tuesday?
Answer: $48.60 First, calculate the price after Monday's loss. The loss was $\frac{1}{10} \times 45 = \$4.50$ . The new price was $\$45 - \$4.50 = \$40.50$ . Next, calculate Tuesday's gain based on this new price. The gain was $\frac{1}{5} \times 40.50 = \$8.10$ . The final price is $\$40.50 + \$8.10 = \$48.60$ .
What is the average (arithmetic mean) of the following three numbers: -4, $\frac{7}{2}$ , and 5.5?
Answer: $1\frac{2}{3}$ To find the average, first sum the numbers. It is easiest to convert all numbers to a common format, like decimals: $-4$ , $3.5$ , and $5.5$ . The sum is $-4 + 3.5 + 5.5 = -4 + 9 = 5$ . Next, divide the sum by the count of the numbers, which is 3. The average is $5 \div 3 = \frac{5}{3}$ . As a mixed number, this is $1\frac{2}{3}$ .
What is the value of the expression $\frac{1}{3} - \left(-\frac{2}{5}\right) \div \left(-\frac{4}{5}\right)$ ?
Answer: $-\frac{1}{6}$ According to the order of operations, perform the division first: $\left(-\frac{2}{5}\right) \div \left(-\frac{4}{5}\right) = \left(-\frac{2}{5}\right) \times \left(-\frac{5}{4}\right) = \frac{10}{20} = \frac{1}{2}$ . Now, substitute this result back into the expression: $\frac{1}{3} - \frac{1}{2}$ . Find a common denominator to subtract: $\frac{2}{6} - \frac{3}{6} = -\frac{1}{6}$ .
Which of the following is equivalent to $\frac{1 - \frac{1}{3}}{2 + \frac{1}{2}}$ ?
Answer: $\frac{4}{15}$ First, simplify the numerator: $1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$ . Next, simplify the denominator: $2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$ . The expression is now a division of two fractions: $\frac{2/3}{5/2}$ . To divide by a fraction, multiply by its reciprocal: $\frac{2}{3} \times \frac{2}{5} = \frac{4}{15}$ .
A length of ribbon $10\frac{1}{2}$ inches long is cut into smaller pieces, each $\frac{3}{4}$ of an inch long. What is the maximum number of full pieces that can be cut, and what is the length of the ribbon left over?
Answer: 14 pieces with 0 inches left over To find how many pieces can be cut, divide the total length by the length of each piece. Convert to improper fractions: $10\frac{1}{2} = \frac{21}{2}$ . The division is $\frac{21}{2} \div \frac{3}{4} = \frac{21}{2} \times \frac{4}{3} = \frac{84}{6} = 14$ . Since the result is an integer, exactly 14 full pieces can be cut with 0 inches left over.

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