AB

7th Grade Math Review

Math Problems

Problem 1: Survey of Science Classes

  • A random sample of 90 students is surveyed from a high school of 900 students.
  • The survey asks each student what science class they are taking.
  • The responses are shown in a table:
    • Biology: 12 students
    • Genetics: 18 students
    • Metallurgy: 20 students
    • Astronomy: 25 students
    • Geology: 15 students
  • The question asks which statement about all the students in the high school is most appropriate based on the survey results.
  • Option A: In a group of 180 students, it is expected that about 36 are taking Metallurgy.
    • This can be checked by calculating the proportion of students taking Metallurgy in the sample (20/90) and applying it to 180 students: (20/90) * 180 = 40. Thus, this option is incorrect.
  • Option B: About 18% of students are taking Genetics.
    • Check by calculating the percentage of students taking genetics: (18/90) * 100 = 20\%. This contradicts the 'about 18%' statement, so this option is incorrect.
  • Option D: Twice as many students are taking Astronomy than are taking Metallurgy.
    • Astronomy has 25 students and Metallurgy has 20. 25 \neq 2 * 20, so this is incorrect. Thus, this option is false
  • Option E: It is estimated that about 150 students are taking Geology.
    • Proportion of geology students: 15/90. Applying this proportion to the entire high school: (15/90) * 900 = 150. This matches the statement, so this option is correct.

Problem 2: Number Line

  • The number line starts at zero.
  • The number line extends to -3 on the left and 2 on the right.
  • The question asks which expression represents the number line.
  • Option A: 2 - (-3). This is equivalent to 2 + 3 = 5, which represents the distance between 2 and -3. This is a possible answer.
  • Option B: (-3) + 2 = -1, which represents the final position after moving 2 units to the right starting at -3. This is not the length of the entire line, so this is incorrect.
  • Option C: 2 + (-3) = -1, same as option B, so incorrect.
  • Option D: (-3) - 2 = -5, moving from -3 by -2, so incorrect.

Problem 3: Equivalent Expressions

  • The equation is 572/1000 = 14.8\%/1000 + x/1000.
  • We need to find x.
  • Multiply both sides by 1000: 572 = 14.8 + x
  • Subtract 14.8 from both sides: x = 572 - 14.8 = 557.2

Problem 4: Spinner Probability

  • The spinner is divided into 8 equal-sized sections.
  • The sections are labeled with numbers: 2, 1, 4, 2, 6, 4, 7, 7.
  • The question asks for the probability of the arrow landing on a section labeled 1 on the first spin.
  • There is one section labeled "1" out of the 8 sections.
  • Therefore, the probability is 1/8.

Problem 5: Area of a Rectangular Bathroom

  • The scale drawing has dimensions 9 inches by 5 inches.
  • The length of the longer side of the actual bathroom is 36 feet.
  • The question asks for the actual area of the bathroom in square feet.
  • First, find the scale factor: \text{scale factor} = \frac{\text{actual length}}{\text{drawing length}} = \frac{36 \text{ feet}}{9 \text{ inches}}
  • Convert 36 feet to inches: 36 \text{ feet} * 12 \text{ inches/foot} = 432 \text{ inches}
  • So, the scale factor is 432/9 = 48.
  • The width of the drawing is 5 inches, so the actual width is 5 * 48 = 240 \text{ inches}.
  • Convert 240 inches to feet: 240 \text{ inches} / 12 \text{ inches/foot} = 20 \text{ feet}.
  • The actual dimensions of the bathroom are 36 feet by 20 feet.
  • Area of actual bathroom = 36 * 20 = 720 \text{ square feet}.

Problem 6: Employee's Hourly Wage

  • The graph shows the relationship between hours worked and total earnings.
  • To determine the hourly wage, find a point on the graph and divide the total earnings by the number of hours.
  • Visually examining the graph: at 4 hours, the pay is \$60.
  • Hourly wage = 60/4 = $15

Problem 7: Wrapping Paper

  • The package dimensions are given as 7 in, 4 in, 4 in, 10 in, and 9 in.
  • The minimum amount of wrapping paper needed is the surface area of the package.
  • The package is a combination of rectangular prisms. Assuming we're looking at a shape that is 7in height with two identical arms of 4 in x 4 in and a top part of 10 in length and 9 in width with the bottom part matching, we need to subtract the overlapping parts of the surface area to avoid double counting the area.
  • Surface Area of the main rectangular prism(without considering the extensions) that has a height of 7, a length of 10, and a width of 9= 2(710 + 79 + 109)= 2(70+63+90)=2(223)=446
  • Surface area of each of the identical perpendicular extensions = 2(44 +47 +47)= 2(16+28+28)=272=144. Since they're two, that's 2*144=288
  • We're subtracting two sides which are each 4 x 4 = 16*2=32
  • Total wrapping paper needed is thus 446+288-32=702 square inches.

Problem 8: Equation Solutions

  • We need to find which equations have x = -44 as a solution.
  • A: -0.25(x - 4) = -15
    • Substitute x = -44: -0.25(-44 - 4) = -0.25(-48) = 12 \neq -15. So, this is incorrect.
  • B: 0.5(x + 5) = 37
    • Substitute x = -44: 0.5(-44 + 5) = 0.5(-39) = -19.5 \neq 37. So this is incorrect.
  • C: 0.875(x + 11) = 28
    • Substitute x = -44: 0.875(-44 + 11) = 0.875(-33) = -28.875 \neq 28. So, this is incorrect.
  • D: -0.5(x - 12) = 28
    • Substitute x = -44: -0.5(-44 - 12) = -0.5(-56) = 28. This is correct.

Problem 9: Volume of Water in Fish Tank

  • Samantha's fish tank holds 17\frac{1}{2} gallons of water when full. Which is 17.5 gallons.
  • She adds 9.6 gallons to fill the tank.
  • The problem asks how many gallons were in the tank before she added water.
  • x + 9.6 = 17.5
  • x = 17.5 - 9.6 = 7.9 \text{ gallons}

Problem 10: Estimating Favorite Subject

  • Declan wants to estimate the number of ninth-grade students whose favorite subject is science.
  • He plans to ask 15 students and wants the sample to be representative.
  • From which population should he randomly select his sample?
  • A: Students at the park.
  • B: Students in a math class.
  • C: Students in a science class.
  • D: Students on a ninth-grade field trip.
  • The best choice is D as this ensures the sample contains only from the ninth grade, whereas using students in a math/science class skews the data.

Problem 11: Circle Area

  • The radius of a circle is 10.3 centimeters.
  • The problem asks for the area of the circle in square centimeters.
  • Area of a circle: A = \pi r^2
  • A = \pi (10.3)^2 = \pi * 106.09 \approx 333.29 \text{ cm}^2

Problem 12: Coin Flip Probability

  • A fair coin is flipped 10 times.
  • The problem asks for the probability of the coin landing on heads all 10 times.
  • The probability of heads on a single flip is 1/2.
  • The probability of 10 heads in a row is (1/2)^{10} = 1/1024

Problem 13: Equivalent Expressions to 140x - 120

  • We need to check if the expressions are equivalent to 140x - 120.
  • A: (700x - 600) / 5 = 140x - 120. This is equivalent.
  • B: (24 - 28x) * 5 = 120 - 140x. This is NOT equivalent. The signs are reversed.
  • C: (28x - 24) * 5 = 140x - 120. This is equivalent.
  • D: (24x - 28) * 5 = 120x - 140. This is NOT equivalent.

Problem 14: Gas Cost Graph

  • The problem contains a graph that shows the relationship between the number of hours (h) a business operates and the total cost (c) of the gas.
  • A: The total cost of gas is \$180 when operating the business for 2 hours.
    • This statement is false as the relation is linear and starts with \$0 for 0 hours. If it was \$180 at 2 hours, it indicates the cost would be \$90 per hour. However, the line doesn't match that scale.
  • B: The total cost of gas is \$4 when operating the business for 360 hours.
    • This implies that after 360 hours, the total cost would be \$4. However, since the relationship is linear, this also doesn't make sense.
  • C: The difference in total cost of gas between point R and point be \$180.
    • The problem is cut off here. Can't assess without knowing those two points.

Problem 15: Bank Transactions

  • Deposits are represented with positive numbers and withdrawals as negative numbers.
  • The question asks what (-3) + 20 represents.
  • A: Three deposits of \$20.
  • B: A \$3 deposit followed by a \$20 withdrawal.
  • C: Three withdrawals of \$20.
  • D: A \$3 withdrawal followed by a \$20 deposit.
  • The expression (-3) + 20 indicates a \$3 withdrawal (represented by -3) followed by a \$20 deposit (represented by +20), therefore, D, is the correct answer.

Problem 16: Proportional Relationship

  • We need to select the table whose ratios of ordered pairs represent a proportional relationship between x and y.
  • A: Not proportional, ratios are not consistent.
  • B: Not proportional, ratios are not consistent.
  • C: Not proportional, ratios are not consistent.
  • D: Ratios must be consistent: 4/5, 12/15 = 4/5, 16/20 = 4/5. This is proportional.

Problem 17: Expression Evaluation

  • The expression is 6 + 12 - (4 / (1/2)) - 4
  • 6 + 12 - (4 * 2) - 4
  • 18 - 8 - 4 = 10 - 4 = 6

Problem 18: Spinner Probability Estimation

  • A spinner has 3 equal sections colored green, yellow, and red.
  • The outcomes of spinning the spinner are given: Green (26), Yellow (35), Red (39).
  • The question asks for the estimated probability of the spinner landing on green.
  • Total outcomes = 26 + 35 + 39 = 100
  • The probability of landing on green is the number of green outcomes divided by the total outcomes:
    • P(\text{Green}) = 26/100 = 0.26

Problem 19: Inequality Solution

  • The inequality is 5x + 3 \le 18
  • Subtract 3 from both sides: 5x \le 15
  • Divide both sides by 5: x \le 3
  • The number line should show a closed circle (or bracket) at 3 and extend to the left (negative infinity).

Problem 20: Expression Evaluation

  • The expression is (5)(-27)(\frac{1}{3})
  • (5)(-27)(\frac{1}{3}) = (5)(-9) = -45

Problem 21: Expected Value

  • The graph shows quantities of sporting goods sold: footballs (81), softballs (66), baseball cleats (48), baseball mitts (30).
  • We need to find the expected number of baseball mitts sold if 675 pieces of sporting equipment were sold.
  • Total items sold = 81+66+48+30 = 225
  • Probability of selling a baseball mitt = 30/225 = 2/15
  • Expected number of baseball mitts sold if 675 pieces of equipment were sold is
    • Expected sales = (2/15) * 675= 90.

Problem 22: Angle Measures in a Circle

  • AD, BE, and CF are diameters of the circle.
  • m\angle AOB = 40^\circ
  • m\angle COE = 150^\circ
  • The problem asks to find the measure of \angle BOC
  • Angles AOB, BOC and COE and are adjacent.
  • The angles AOB, BOC, COE form a line, thus their angles should sum to 180 degrees.
  • Thus, AOB + BOC + COE + = 180.
  • BOC = 180 - AOB - COE= 180 - 40 - 150 = -10. This does not seem right- need to look into problem further. Assuming the angle is reflex one needs to realize that angles around point=360. It may be AOD + DOC, thus AOB + 180-COE, for some reason the diagram is not correct.

Problem 23: Experimental Probability

  • A table shows the results of randomly selecting colored marbles from a bag 31 times.
    • Orange: 2
    • Green: 8
    • Red: 1
    • Pink: 7
    • Black: 3
    • Blue: 10
  • We want to find the expected probability of selecting a pink marble from the bag in one attempt.
  • The total number of selections is 31.
  • The number of times pink was selected is 7.
  • The experimental probability of selecting pink is 7/31

Problem 24: Constant of Proportionality

  • The linear graph shown is used to determine the cost of a cell phone plan.
  • We need to determine the constant of proportionality (p) and enter an equation in the form of b = pn.
  • Without the information of the graph it's not possible to derive the constant of proportionality. Assuming the graph implies for N=1 the constant is say- 20 we've b=20n.