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= $2<em>(7</em>10 + 7<em>9 + 10</em>9)= 2<em>(70+63+90)=2</em>(223)=446$
• Surface area of each of the identical perpendicular extensions = $2<em>(4</em>4 +4<em>7 +4</em>7)= 2<em>(16+28+28)=2</em>72=144$. 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.