Solving for an Unknown - ISEE Middle Level Mathematics Achievement

What is the value of $p$ that satisfies the equation $14 - (2p - 5) = 3$ ?
Answer: 8 To solve the equation, first distribute the negative sign to both terms inside the parentheses: $14 - 2p + 5 = 3$ . Combine the constant terms on the left side: $19 - 2p = 3$ . Subtract 19 from both sides: $-2p = 3 - 19$ , which gives $-2p = -16$ . Finally, divide both sides by -2 to find $p = 8$ .
For what value of $k$ is $x = 5$ a solution to the equation $3(x - 2) + k = 4x - 1$ ?
Answer: 10 To find the value of $k$ , substitute $x = 5$ into the equation. The equation becomes $3(5 - 2) + k = 4(5) - 1$ . Simplify both sides: $3(3) + k = 20 - 1$ , which is $9 + k = 19$ . To solve for $k$ , subtract 9 from both sides: $k = 19 - 9 = 10$ .
At a carnival, ride tickets cost $2.00 each. The entrance fee is $8.00. If Marco spent a total of $34.00, how many ride tickets did he buy?
Answer: 13 Let $t$ be the number of ride tickets Marco bought. The total cost is the sum of the entrance fee and the cost of the tickets. The equation is $2t + 8 = 34$ . First, subtract the $8.00 entrance fee from the total amount spent: $2t = 34 - 8$ , which is $2t = 26$ . Then, divide the remaining amount by the cost per ticket to find the number of tickets: $t = 26 / 2 = 13$ . Marco bought 13 tickets.
A rectangle's length is 1 centimeter less than twice its width. If the perimeter of the rectangle is 52 centimeters, what is its width in centimeters?
Answer: 9 Let $w$ be the width. The length $l$ is $2w - 1$ . The formula for the perimeter of a rectangle is $P = 2l + 2w$ . Substitute the given values: $52 = 2(2w - 1) + 2w$ . Distribute the 2: $52 = 4w - 2 + 2w$ . Combine like terms: $52 = 6w - 2$ . Add 2 to both sides: $54 = 6w$ . Divide by 6 to find $w = 9$ . The width is 9 cm.
A water tank is $\frac{1}{4}$ full. After adding 10 gallons of water, the tank is $\frac{2}{3}$ full. What is the total capacity of the tank in gallons?
Answer: 24 Let $C$ be the total capacity of the tank. The initial amount of water is $\frac{1}{4}C$ . After adding 10 gallons, the amount is $\frac{1}{4}C + 10$ , which is equal to $\frac{2}{3}C$ . The equation is $\frac{1}{4}C + 10 = \frac{2}{3}C$ . To solve for $C$ , subtract $\frac{1}{4}C$ from both sides: $10 = \frac{2}{3}C - \frac{1}{4}C$ . Find a common denominator (12): $10 = \frac{8}{12}C - \frac{3}{12}C$ , which simplifies to $10 = \frac{5}{12}C$ . Multiply by $\frac{12}{5}$ to isolate $C$ : $C = 10 \times \frac{12}{5} = \frac{120}{5} = 24$ . The capacity is 24 gallons.
A company has 45 employees. The number of employees in the sales department is one more than twice the number of employees in the production department. All other 8 employees are in administration. How many employees are in the sales department?
Answer: 25 Let $s$ be the number of sales employees and $p$ be the number of production employees. The total number of employees is 45. There are 8 in administration, so $s + p + 8 = 45$ . This means $s + p = 37$ . We are also told that $s = 2p + 1$ . Substitute this into the previous equation: $(2p + 1) + p = 37$ . Combine like terms: $3p + 1 = 37$ . Subtract 1: $3p = 36$ . Divide by 3: $p = 12$ . There are 12 employees in production. To find the number of sales employees, use $s = 2p + 1$ : $s = 2(12) + 1 = 24 + 1 = 25$ . There are 25 employees in the sales department.
A runner completes 24 miles over 4 days, running the same distance each day. No rest days are included in the total. Let $x$ be the miles run each day. The situation is modeled by $4x = 24$ . If $4x = 24$ , what is $x$ ?
Answer: $x = 6$ This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 4x = 24, where the unknown variable is isolated by dividing both sides by 4 to find x = 6. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 4 * 7 = 28. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.
If $4n - 5 = 11$ , what is the value of the expression $8n + 3$ ?
Answer: 35 First, solve the given equation for $n$ . Add 5 to both sides: $4n = 11 + 5$ , which simplifies to $4n = 16$ . Divide by 4 to find $n = 4$ . Next, substitute this value of $n$ into the expression $8n + 3$ . This gives $8(4) + 3 = 32 + 3 = 35$ .
A bus travels 150 miles in 3 hours at a constant speed. The driver keeps the same speed for the whole trip. Let $x$ be the speed in miles per hour. The situation can be written as $3x = 150$ . If $3x = 150$ , what is $x$ ?
Answer: $x = 50$ This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x = 150, where the unknown variable is isolated by dividing both sides by 3 to find x = 50. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 3 * 150 = 450. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.
If $\frac{3x}{4} + 5 = 11$ , what is the value of $x - 2$ ?
Answer: 6 First, solve the equation for $x$ . Subtract 5 from both sides: $\frac{3x}{4} = 11 - 5$ , which is $\frac{3x}{4} = 6$ . Multiply both sides by 4: $3x = 24$ . Divide by 3: $x = 8$ . The question asks for the value of $x - 2$ . Substitute the value of $x$ : $8 - 2 = 6$ .
What is the solution to the equation $2(3x - 4) = 7(x - 1)$ ?
Answer: -1 First, distribute on both sides of the equation: $6x - 8 = 7x - 7$ . Next, subtract $6x$ from both sides to get $-8 = x - 7$ . Finally, add 7 to both sides to isolate $x$ : $-8 + 7 = x$ , which simplifies to $x = -1$ .
Maya buys 3 notebooks and 1 binder, and the total is $18. The binder costs $6. The notebooks all cost the same amount, and sales tax is already included. The situation can be modeled by $3x + 6 = 18$ , where $x$ is the price of one notebook. If $3x + 6 = 18$ , what is $x$ ?
Answer: $x = 4$ This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x + 6 = 18, where the unknown variable is isolated by subtracting 6 from both sides to get 3x = 12 and then dividing by 3 to find x = 4. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students add 6 instead of subtracting, leading to 3x = 24 and x = 8. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.
Sam is 4 years younger than twice his sister's age. The sum of their ages is 26. How old is Sam?
Answer: 16 Let Sam's age be $S$ and his sister's age be $R$ . From the problem, we can write two equations: $S = 2R - 4$ and $S + R = 26$ . Substitute the first equation into the second: $(2R - 4) + R = 26$ . Combine like terms: $3R - 4 = 26$ . Add 4 to both sides: $3R = 30$ . Divide by 3 to find the sister's age, $R = 10$ . Now find Sam's age using $S = 2R - 4$ : $S = 2(10) - 4 = 20 - 4 = 16$ . Sam is 16 years old.
What is the value of $z$ in the equation $\frac{36}{z+1} = 4$ ?
Answer: 8 To solve for $z$ , you can first multiply both sides of the equation by $z+1$ to get $36 = 4(z+1)$ . Then, divide both sides by 4 to get $9 = z+1$ . Finally, subtract 1 from both sides to find $z = 8$ . Alternatively, recognize that if $36$ divided by a number is 4, that number must be 9. So, $z+1=9$ , which means $z=8$ .
At a certain store, the price of a bulk grain is directly proportional to its weight. If 3.5 pounds of the grain cost $8.40, what is the cost of 5 pounds of the same grain?
Answer: $12.00 First, find the price per pound by dividing the total cost by the weight: $\$8.40 / 3.5$ pounds. This can be calculated as $840 / 350 = 84 / 35 = 12 / 5 = 2.4$ . So, the grain costs $2.40 per pound. To find the cost of 5 pounds, multiply the unit price by the weight: $5 \times \$2.40 = \$12.00$ .
What is the solution to the equation $7x - 8 = 3x + 12$ ?
Answer: 5 To solve for $x$ , first gather the variable terms on one side of the equation. Subtract $3x$ from both sides: $7x - 3x - 8 = 12$ , which simplifies to $4x - 8 = 12$ . Next, add 8 to both sides: $4x = 12 + 8$ , which gives $4x = 20$ . Finally, divide both sides by 4 to get $x = 5$ .
What is the value of $m$ in the equation $\frac{2}{3}m - 1 = \frac{1}{2}m + 2$ ?
Answer: 18 To solve this equation, first eliminate the fractions by multiplying both sides by the least common denominator of 3 and 2, which is 6. This gives $6(\frac{2}{3}m - 1) = 6(\frac{1}{2}m + 2)$ , which simplifies to $4m - 6 = 3m + 12$ . Subtract (3m) from both sides to get $m - 6 = 12$ . Add 6 to both sides to find $m = 18$ .
A baker makes 5 identical loaves and sells them for $40 total. Each loaf has the same price, and there are no coupons. Let $x$ be the price of one loaf. The situation is represented by $5x = 40$ . Determine the value of $x$ when $5x = 40$ .
Answer: $x = 8$ This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 5x = 40, where the unknown variable is isolated by dividing both sides by 5 to find x = 8. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 5 * 40 = 200. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.
Solve for $y$ : $5(y - 3) = 3y + 1$
Answer: 8 First, use the distributive property on the left side: $5y - 15 = 3y + 1$ . Next, subtract $3y$ from both sides to get $2y - 15 = 1$ . Then, add 15 to both sides: $2y = 16$ . Finally, divide by 2 to find $y = 8$ .
A car travels at a constant speed for 3 hours and goes 180 miles. The driver does not stop, and the distance stays the same the whole time. Let $x$ be the car's speed in miles per hour. The situation can be written as $3x = 180$ . Solve for $x$ in the equation $3x = 180$ .
Answer: $x = 60$ This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 3x = 180, where the unknown variable is isolated by dividing both sides by 3 to find x = 60. The correct answer, choice A, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice B is incorrect due to a common arithmetic error where students multiply instead of dividing, such as 3 * 180 = 540. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.

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