Equations with Fractions & Decimals - Accuplacer Qas

If $m$ is a number such that $\frac{1}{5}m + \frac{2}{3}m = 13$ , what is the value of $m$ ?
Answer: $15$ To solve for $m$ , first combine the terms on the left side. The least common denominator for 5 and 3 is 15. Rewrite the fractions: $\frac{3}{15}m + \frac{10}{15}m = 13$ . Combine them: $\frac{13}{15}m = 13$ . To isolate $m$ , you can divide both sides by 13, which gives $\frac{1}{15}m = 1$ . Then, multiply both sides by 15 to get $m = 15$ .
A student's final grade in a course is calculated as $\frac{3}{5}$ of the test average plus $\frac{2}{5}$ of the final exam score. A student has a test average of 85. If the student's final grade was 88, what was their final exam score?
Answer: $92.5$ Let $E$ be the final exam score. The formula for the final grade is $G = \frac{3}{5}(Test Average) + \frac{2}{5}(E)$ . We are given $G = 88$ and Test Average = 85. Substitute these values into the equation: $88 = \frac{3}{5}(85) + \frac{2}{5}E$ . First, calculate $\frac{3}{5}(85)$ : $3 \times \frac{85}{5} = 3 \times 17 = 51$ . The equation becomes $88 = 51 + \frac{2}{5}E$ . Subtract 51 from both sides: $37 = \frac{2}{5}E$ . To solve for $E$ , multiply both sides by $\frac{5}{2}$ : $E = 37 \times \frac{5}{2} = \frac{185}{2} = 92.5$ .
A solution contains $0.3x + 1.7 = 2.6x - 0.8$ . After converting to eliminate decimals by multiplying by 10, what equation must be solved?
Answer: $3x + 17 = 26x - 8$ Multiplying the entire equation $0.3x + 1.7 = 2.6x - 0.8$ by 10 gives: $10(0.3x) + 10(1.7) = 10(2.6x) - 10(0.8)$ , which becomes $3x + 17 = 26x - 8$ . Choice B incorrectly multiplies 0.3 as 30, Choice C multiplies some terms by 100, Choice D incorrectly makes 2.6x into 260x.
At a certain company, $\frac{1}{3}$ of the employees are in sales and $\frac{1}{4}$ are in management. The remaining 35 employees are in production. How many employees work at the company in total?
Answer: $84$ Let T be the total number of employees. The fraction of employees in sales and management combined is $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$ . The remaining fraction of employees, who are in production, is $1 - \frac{7}{12} = \frac{5}{12}$ . We are given that these remaining employees number 35. So, $\frac{5}{12}T = 35$ . To find T, multiply both sides by $\frac{12}{5}$ : $T = 35 \times \frac{12}{5} = 7 \times 12 = 84$ .
What is the solution to the equation $\frac{z-2}{6} + \frac{z}{4} = 5$ ?
Answer: $12.8$ The least common denominator of 6 and 4 is 12. Multiply every term in the equation by 12 to clear the fractions: $12(\frac{z-2}{6}) + 12(\frac{z}{4}) = 12(5)$ . This simplifies to $2(z-2) + 3z = 60$ . Distribute the 2: $2z - 4 + 3z = 60$ . Combine the $z$ terms: $5z - 4 = 60$ . Add 4 to both sides: $5z = 64$ . Divide by 5: $z = \frac{64}{5} = 12.8$ .
What is the solution to the equation $\frac{2}{3}(x - 6) = \frac{1}{2}x$ ?
Answer: $24$ To solve the equation $\frac{2}{3}(x - 6) = \frac{1}{2}x$ , first distribute $\frac{2}{3}$ on the left side: $\frac{2}{3}x - \frac{12}{3} = \frac{1}{2}x$ , which simplifies to $\frac{2}{3}x - 4 = \frac{1}{2}x$ . To eliminate the fractions, multiply all terms by the least common denominator, which is 6. This gives $6(\frac{2}{3}x) - 6(4) = 6(\frac{1}{2}x)$ , which simplifies to $4x - 24 = 3x$ . Subtracting $3x$ from both sides gives $x - 24 = 0$ . Adding 24 to both sides gives $x = 24$ .
What is the solution to the equation $\frac{3}{4}x - 0.5 = 0.25x$ ?
Answer: $1$ This equation contains both a fraction and decimals. It's often easiest to convert everything to one form. Converting to decimals: $\frac{3}{4}$ is 0.75, so the equation becomes $0.75x - 0.5 = 0.25x$ . Subtract $0.25x$ from both sides: $0.50x - 0.5 = 0$ . Add 0.5 to both sides: $0.50x = 0.5$ . Divide by 0.5: $x = 1$ . Alternatively, converting to fractions: 0.5 is $\frac{1}{2}$ and 0.25 is $\frac{1}{4}$ . The equation becomes $\frac{3}{4}x - \frac{1}{2} = \frac{1}{4}x$ . Multiply all terms by the LCD, 4, to get $3x - 2 = x$ . Subtract $x$ from both sides: $2x - 2 = 0$ . Add 2 to both sides: $2x = 2$ . Divide by 2: $x = 1$ .
The average (arithmetic mean) of three numbers is 15. The first number is $\frac{1}{2}x$ . The second number is $\frac{1}{3}x$ . The third number is 10. What is the value of $x$ ?
Answer: $42$ The average of three numbers is their sum divided by 3. The equation is $\frac{\frac{1}{2}x + \frac{1}{3}x + 10}{3} = 15$ . First, multiply both sides by 3 to get the sum: $\frac{1}{2}x + \frac{1}{3}x + 10 = 45$ . Subtract 10 from both sides: $\frac{1}{2}x + \frac{1}{3}x = 35$ . To combine the fractions, find a common denominator (6): $\frac{3}{6}x + \frac{2}{6}x = 35$ , which is $\frac{5}{6}x = 35$ . To solve for $x$ , multiply both sides by the reciprocal $\frac{6}{5}$ : $x = 35 \times \frac{6}{5} = 7 \times 6 = 42$ .
What value of $p$ satisfies the equation $\frac{4p - 2}{p} = 3$ ?
Answer: $2$ Assuming $p \neq 0$ , we can solve this equation by multiplying both sides by $p$ to clear the denominator. This gives $4p - 2 = 3p$ . To isolate the variable $p$ , subtract $3p$ from both sides: $p - 2 = 0$ . Finally, add 2 to both sides to find $p = 2$ .
A customer's monthly phone bill is calculated as a flat fee of $15.00 plus $2.50 per gigabyte of data used. If a customer's bill for one month was $52.50, how many gigabytes of data did they use?
Answer: $15$ Let $g$ be the number of gigabytes of data used. The total cost can be represented by the equation $15.00 + 2.50g = 52.50$ . To find $g$ , first subtract the flat fee from the total bill: $2.50g = 52.50 - 15.00$ , which simplifies to $2.50g = 37.50$ . Then, divide the remaining cost by the price per gigabyte: $g = \frac{37.50}{2.50} = 15$ . The customer used 15 gigabytes of data.
What value of $y$ satisfies the equation $\frac{3}{5}y - 2 = \frac{1}{3}y + 6$ ?
Answer: $30$ To solve the equation, first eliminate the fractions by multiplying every term by the least common denominator of 5 and 3, which is 15. The equation becomes $15(\frac{3}{5}y) - 15(2) = 15(\frac{1}{3}y) + 15(6)$ , which simplifies to $9y - 30 = 5y + 90$ . Next, gather the variable terms on one side by subtracting $5y$ from both sides: $4y - 30 = 90$ . Add 30 to both sides: $4y = 120$ . Finally, divide by 4: $y = 30$ .
A rectangle has a length of $\frac{1}{2}x + 3$ units and a width of $\frac{1}{3}x - 1$ units. If the perimeter of the rectangle is 34 units, what is the value of $x$ ?
Answer: $18$ The formula for the perimeter of a rectangle is $P = 2(L+W)$ . Substitute the given expressions: $34 = 2((\frac{1}{2}x + 3) + (\frac{1}{3}x - 1))$ . First, simplify the expression inside the parentheses by combining like terms: $\frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x$ , and $3 - 1 = 2$ . The equation becomes $34 = 2(\frac{5}{6}x + 2)$ . Distribute the 2: $34 = \frac{10}{6}x + 4$ , which simplifies to $34 = \frac{5}{3}x + 4$ . Subtract 4 from both sides: $30 = \frac{5}{3}x$ . To solve for $x$ , multiply both sides by $\frac{3}{5}$ : $x = 30 \times \frac{3}{5} = 18$ .
The equation $0.2(y - 5) + 0.3 = 1.7$ is given. What is the value of $y$ ?
Answer: $12$ First, isolate the term containing the parentheses by subtracting 0.3 from both sides: $0.2(y - 5) = 1.7 - 0.3$ , which simplifies to $0.2(y - 5) = 1.4$ . Next, divide both sides by 0.2: $y - 5 = \frac{1.4}{0.2} = 7$ . Finally, add 5 to both sides to solve for $y$ : $y = 7 + 5 = 12$ .
Solve for $z$ : $\frac{z + 5}{3} = \frac{z - 1}{2}$
Answer: $13$ To solve this proportion, you can cross-multiply. This gives $2(z + 5) = 3(z - 1)$ . Distribute the numbers on both sides: $2z + 10 = 3z - 3$ . To solve for $z$ , gather the $z$ terms on one side and the constants on the other. Subtract $2z$ from both sides: $10 = z - 3$ . Add 3 to both sides: $13 = z$ .
A chemist mixes a quantity of a 20% acid solution with a 60% acid solution to produce 50 liters of a 30% acid solution. How many liters of the 20% acid solution were used?
Answer: $37.5$ Let $x$ be the number of liters of the 20% solution. Then $50 - x$ is the number of liters of the 60% solution. The total amount of acid in the final mixture is the sum of the amounts of acid from each solution. This can be written as the equation: $0.20x + 0.60(50 - x) = 0.30(50)$ . Distribute on the left side: $0.20x + 30 - 0.60x = 15$ . Combine the $x$ terms: $-0.40x + 30 = 15$ . Subtract 30 from both sides: $-0.40x = -15$ . Divide by -0.40: $x = \frac{-15}{-0.40} = 37.5$ . So, 37.5 liters of the 20% solution were used.
If $4.5 + 0.2(x-1) = 6$ , what is the value of $x$ ?
Answer: $8.5$ To solve for $x$ , first isolate the term containing the variable. Subtract 4.5 from both sides of the equation: $0.2(x-1) = 6 - 4.5$ , which simplifies to $0.2(x-1) = 1.5$ . Next, divide both sides by 0.2: $x-1 = \frac{1.5}{0.2} = 7.5$ . Finally, add 1 to both sides: $x = 7.5 + 1 = 8.5$ .
If $\frac{2}{3}k - 4 = 2$ , what is the value of the expression $k - 3$ ?
Answer: $6$ First, solve the given equation for $k$ . Add 4 to both sides of $\frac{2}{3}k - 4 = 2$ to get $\frac{2}{3}k = 6$ . To isolate $k$ , multiply both sides by the reciprocal of $\frac{2}{3}$ , which is $\frac{3}{2}$ . This gives $k = 6 \times \frac{3}{2} = 9$ . The question asks for the value of $k - 3$ . Substitute the value of $k$ : $9 - 3 = 6$ .
If $\frac{2}{3}x + \frac{5}{8} = \frac{3}{4}x - \frac{1}{6}$ , what is the value of $x$ ?
Answer: $x = \frac{19}{2}$ Rearrange to get all $x$ terms on one side: $\frac{2}{3}x - \frac{3}{4}x = -\frac{1}{6} - \frac{5}{8}$ . Find common denominators: $\frac{8}{12}x - \frac{9}{12}x = -\frac{4}{24} - \frac{15}{24}$ . This simplifies to $-\frac{1}{12}x = -\frac{19}{24}$ . Multiply both sides by $-12$ : $x = \frac{19 \times 12}{24} = \frac{19}{2}$ . Choice B incorrectly uses 5 in denominator, Choice C uses wrong arithmetic, Choice D makes fraction errors.
If $0.4(x + 10) - 0.1x = 1$ , what is the value of $x$ ?
Answer: $-10$ To solve for $x$ , first distribute the 0.4: $0.4x + 4 - 0.1x = 1$ . Combine the terms with $x$ : $0.3x + 4 = 1$ . Subtract 4 from both sides: $0.3x = -3$ . Finally, divide by 0.3: $x = \frac{-3}{0.3} = -10$ . Alternatively, multiply the entire equation by 10 to clear the decimals: $4(x + 10) - 1x = 10$ , which becomes $4x + 40 - x = 10$ , then $3x + 40 = 10$ , so $3x = -30$ , and $x = -10$ .
The equation $1.2x - 0.4(x + 3) = 0.6x + 1.8$ has what solution?
Answer: $x = 15$ Expand: $1.2x - 0.4x - 1.2 = 0.6x + 1.8$ . Combine like terms: $0.8x - 1.2 = 0.6x + 1.8$ . Subtract $0.6x$ and add $1.2$ : $0.2x = 3$ . Therefore $x = 15$ . Choice A results from decimal calculation errors, Choice C from incorrectly combining terms, Choice D incorrectly assumes the equation has no solution when coefficients appear similar.

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