In a triangle, the measure of the second angle is twice the measure of the first angle. The measure of the third angle is 20 degrees more than the first angle. What is the measure of the largest angle?
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ISEE Middle Level Mathematics Achievement Quiz
Practice Solving For An Unknown in ISEE Middle Level Mathematics Achievement with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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In a triangle, the measure of the second angle is twice the measure of the first angle. The measure of the third angle is 20 degrees more than the first angle. What is the measure of the largest angle?
This quiz focuses on Solving For An Unknown, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
In a triangle, the measure of the second angle is twice the measure of the first angle. The measure of the third angle is 20 degrees more than the first angle. What is the measure of the largest angle?
Explanation: Let the measure of the first angle be (x). Then the second angle is (2x) and the third angle is (x + 20). The sum of the angles in a triangle is 180 degrees. So, (x + 2x + (x + 20) = 180). Combine like terms: (4x + 20 = 180). Subtract 20 from both sides: (4x = 160). Divide by 4: (x = 40). The angles are (x=40) degrees, (2x = 2(40) = 80) degrees, and (x + 20 = 40 + 20 = 60) degrees. The largest of these is 80 degrees.
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?
Explanation: 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.
What is the solution to the equation (2(3x - 4) = 7(x - 1))?
Explanation: 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).
At a carnival, ride tickets cost 2.00each.Theentrancefeeis8.00. If Marco spent a total of $34.00, how many ride tickets did he buy?
Explanation: 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 $8.00 entrance fee from the total amount spent: (2t = 34 - 8), which is (2t = 26). Then, divide the remaining amount by the cost per ticket to find the number of tickets: (t = 26 / 2 = 13). Marco bought 13 tickets.
A water tank is (\frac{1}{4}) full. After adding 10 gallons of water, the tank is (\frac{2}{3}) full. What is the total capacity of the tank in gallons?
Explanation: Let (C) be the total capacity of the tank. The initial amount of water is (\frac{1}{4}C). After adding 10 gallons, the amount is (\frac{1}{4}C + 10), which is equal to (\frac{2}{3}C). The equation is (\frac{1}{4}C + 10 = \frac{2}{3}C). To solve for (C), subtract (\frac{1}{4}C) from both sides: (10 = \frac{2}{3}C - \frac{1}{4}C). Find a common denominator (12): (10 = \frac{8}{12}C - \frac{3}{12}C), which simplifies to (10 = \frac{5}{12}C). Multiply by (\frac{12}{5}) to isolate (C): (C = 10 \times \frac{12}{5} = \frac{120}{5} = 24). The capacity is 24 gallons.
For what value of (k) is (x = 5) a solution to the equation (3(x - 2) + k = 4x - 1)?
Explanation: To find the value of (k), substitute (x = 5) into the equation. The equation becomes (3(5 - 2) + k = 4(5) - 1). Simplify both sides: (3(3) + k = 20 - 1), which is (9 + k = 19). To solve for (k), subtract 9 from both sides: (k = 19 - 9 = 10).
A streaming service charges 12eachmonthplusa5 one-time fee. After a few months, the total cost is 53.Letxbethenumberofmonthspaidfor.Thissituationisrepresentedby12x + 5 = 53.If12x + 5 = 53,whatisx$?
Explanation: 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 12x + 5 = 53, where the unknown variable is isolated by subtracting 5 from both sides to get 12x = 48 and then dividing by 12 to find x = 4. 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 forget to subtract and divide 53 by something to get x = 48. 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.
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?
Explanation: 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.
A recipe uses 2 cups of flour for 8 muffins. Sam wants to make 24 muffins and keeps the same recipe ratio. Let x be the number of cups of flour needed. The situation can be written as 8x=48. Solve for x in the equation 8x=48.
Explanation: 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 8x = 48, where the unknown variable is isolated by dividing both sides by 8 to find x = 6. 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 8 * 5 = 40. 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.
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?
Explanation: 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.
A class orders 9 identical pizzas for a total of 99.Eachpizzacoststhesameamount,anddeliveryisfree.Letxbethecostofonepizza.Thesituationisrepresentedby9x = 99.Solveforxintheequation9x = 99$.
Explanation: This question tests middle school mathematics skills: solving linear equations for an unknown variable. To solve for an unknown, isolate the variable by performing inverse operations—undo addition with subtraction, multiplication with division, etc. In this scenario, the equation presented is 9x = 99, where the unknown variable is isolated by dividing both sides by 9 to find x = 11. The correct answer, choice C, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice A is incorrect due to a common arithmetic error where students divide incorrectly, such as 99 / 11 = 9. To help students master this skill, practice breaking down equations step-by-step, checking each inverse operation. Encourage identifying operations by highlighting terms that need isolation and using manipulatives or visual aids to reinforce understanding.
A taxi charges a 4startfeeplus2 per mile. Jordan’s ride costs 18total,includingthestartfee.Letxbethenumberofmilestraveled.Thissituationisrepresentedby2x + 4 = 18.If2x + 4 = 18,whatisx$?
Explanation: 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 2x + 4 = 18, where the unknown variable is isolated by subtracting 4 from both sides to get 2x = 14 and then dividing by 2 to find x = 7. 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 subtract incorrectly, such as 18 - 4 = 14 but then divide by something else leading to x = 11. 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.
A bookstore sells a novel for xdollars,andabookmarkcosts2. Ava buys 4 novels and 1 bookmark for 34total.Allnovelshavethesameprice.Thesituationismodeledby4x + 2 = 34.Findxif4x + 2 = 34$.
Explanation: 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 + 2 = 34, where the unknown variable is isolated by subtracting 2 from both sides to get 4x = 32 and then dividing by 4 to find x = 8. The correct answer, choice B, reflects the accurate isolation and calculation of the variable, ensuring the equation balances. Choice C is incorrect due to a common arithmetic error where students forget to subtract and divide 34 by something to get x = 32. 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?
Explanation: 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.
The mean of five numbers is 16. Four of the numbers are 12, 18, 7, and 23. What is the fifth number?
Explanation: The mean of a set of numbers is the sum of the numbers divided by the count of the numbers. The sum of the five numbers is the mean multiplied by the count: (16 \times 5 = 80). The sum of the four known numbers is (12 + 18 + 7 + 23 = 60). To find the fifth number, subtract the sum of the four numbers from the total sum: (80 - 60 = 20). The fifth number is 20.
If (\frac{3x}{4} + 5 = 11), what is the value of (x - 2)?
Explanation: 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).
If (4n - 5 = 11), what is the value of the expression (8n + 3)?
Explanation: 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).
What is the value of (m) in the equation (\frac{2}{3}m - 1 = \frac{1}{2}m + 2)?
Explanation: 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 40total.Eachloafhasthesameprice,andtherearenocoupons.Letxbethepriceofoneloaf.Thesituationisrepresentedby5x = 40.Determinethevalueofxwhen5x = 40$.
Explanation: 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.
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.
Explanation: 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.