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SAT Math Test 10 Section 4 solved questions 1 to 20

Disclaimer: Please note that this is not a tutorial. Solutions to questions are shown in a step-wise approach using mathematical concepts. Students viewing these answers may or may not be familiar with these concepts. Each question mentions the Category in my book that prepares the student on the concepts.
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SAT Practice Test 10 Section 4 Questions 1 to 20 Answer Explanations

Step 1: Identify the components of the equation
Initial altitude = 40 feet.
Increase in altitude = 21 feet per second. Hence, altitude increase in 𝑡 seconds = 21𝑡.
Altitude in 𝑡 seconds = 𝑦.
Step 2: Determine the equation
altitude in 𝑡 seconds = initial altitude + altitude increase in 𝑡 seconds
𝑦 = 40 + 21𝑡
Maps to Category 14 – Word problems on linear equations with one variable.

Step 1: Match the sequence of events in the question with the flow of events in the graph
𝑥-axis = number of texts.
𝑦-axis = cost of texts.
The plan charges a flat fee of \$5 for up to 100 texts in a month. Hence, there is no additional cost for the first 100 texts in a month. This will be represented by a horizontal line starting at 𝑦 (cost) = 5 and extending until 𝑥 (texts) = 100. This eliminates answer choices C and D.
Since the cost goes up after 100 texts, the line will go upwards in the direction of increasing cost. This eliminates answer choice B.
Maps to Category 53 – Graphs with line segments and curves.

Step 1: Match the sequence of events in the question with the flow of events in the graph
𝑥-axis = time in minutes.
𝑦-axis = amount of popcorn.
Since Jake ate half the popcorns in 15 minutes and then finished the remaining after waiting for 30 minutes, the amount of popcorns cannot increase (go up the 𝑦-axis) at any time point. This eliminates answer choices A, C, and D.
Maps to Category 53 – Graphs with line segments and curves.

Step 1: Isolate variable on one side of the equation and solve
20 – 𝑥 = 15      𝑥 = 20 – 15 = 5
Step 2: Determine 3𝑥
3𝑥 = 3 × 5 = 15
There are several categories that demonstrate how to solve for a variable in an equation.

Step 1: Plug in the given value of 𝑥 in the equation
In 𝑓(1), 1 is the given value of 𝑥
$$f(-1)=\frac {-1+3}{2}=\frac{2}{2}=1$$
Maps to Category 21 – Polynomial functions as equations.

Step 1: Simplify the expression
Multiply each term within the parenthesis with 2𝑥.
$$2x(x^2-3x)={(2x\times x^2)}-{(2x\times 3x)}={2x^3}-{6x^2}$$
There are several categories that demonstrate how to simplify an expression.

A sample should be a random sample from the entire sample population. This eliminates answer choices A, C, and D. In answer choice A, the sample is restricted to one store. In answer choice C, the sample is restricted by employee pay. Answer choice D does not represent a true survey.
Maps to Category 65 – Interpretation of sample data in studies and surveys.

Step 1: Read the graph to determine the weekly deposit for Ian
Deposit at week 0 = 100.
Deposit at week 7 = 800.
deposit in 7 weeks = 800 – 100 = 700
weekly deposit is 700 divided by 7  = 100
Step 2: Read the graph to determine the weekly deposit for Jeremy
Deposit at week 0 = 300.
Deposit at week 7 = 650.
deposit in 7 weeks = 650 – 300 = 350
weekly deposit is 350 divided by 7  = 50
Step 3: Determine the difference
100 – 50 = 50
Maps to Category 53 – Graphs with line segments and curves.

Step 1: Plug in the given value of 𝑥 in the equation
In 𝘩(5), 5 is the given value of 𝑥 and in 𝘩(3), 3 is the given value of 𝑥.
$$h(5)-h(3)=2^5-2^3=32-8=24$$
Maps to Category 21 – Polynomial functions as equations.

Margin of error is the plausibility/probability by which the results of a survey or a study are expected to differ. It is calculated as the positive and the negative deviation of the results. In this question, the margin of error is between 23 – 4 = 19 and 23 + 4 = 27. This eliminates answer choices A and B. Answer choice C is incorrect because margin of error is a plausibility not a guarantee.
Maps to Category 65 – Interpretation of sample data in studies and surveys.

Step 1: Evaluate the standard deviation
Standard deviation is the spread of the numbers in a data set.
In List A, the numbers are spread between 1 and 6.
In List B, the numbers are spread between 2 and 5.
Hence, the spread is different. This eliminates answer choices B and D.
Step 2: Determine the mean
Mean of List A is
$$1+2 +3 +4 + 5 +6=\frac {21}{6}=3.5$$
Mean of List B is
$$2+3+3+4+4+5=\frac {21}{6}=3.5$$
Means are same. This eliminates answer choice C.
Maps to Category 64 – Comparing mean, median, mode, range, and SD.

Step 1: Determine the percent decrease from original price to sale price as a decimal
percent decrease in price as decimal = 1 – 0.4 = 0.6
Step 2: Determine the original price
Sale price = 18.
$$\frac {18}{0.6}=30$$
Maps to Category 46 – The original number before a percent increase/decrease.

Step 1: Evaluate the height of the bars
The height of a bar represents the number of insects in a colony. It can be observed from the graph that for Colony C the height of the bar increased after 2 weeks of treatment. This eliminates Colony C. For Colony A and Colony B, the height of the bars show continuous decrease. Hence, options I and II are correct.
Maps to Category 55 – Bar graphs.

Step 1: Read the height of all the bars at week 0
Time of initial treatment is week 0. The total number of insects at week 0 is the sum of the height of all the three bars at week 0. If the bars do not end at graph gridlines, then read approximate height.
Colony A + Colony B + Colony C = 80 + 65 + 58 = 203 approximately
Step 2: Read the height of all the bars at week 8
The total number of insects at week 8 is the sum of the height of all the three bars at week 8.
Colony A + Colony B + Colony C = 18 + 10 + 50 = 78 approximately
Step 3: Determine the ratio
Since the question asks for the closest ratio, approximate the numbers.
week 8:week 0 = 78:203 =  80:200 = 2:5
Maps to Category 55 – Bar graphs.

Step 1: Determine the radius of the cone
Formula for the volume 𝑉 of a cone is
$$V={\frac {1}{3}}{\pi}{r^2}h$$
The volume 𝑉 and the height 𝘩 are given in the question. Plug these values in the formula and determine the radius 𝑟.
$${24\pi}={\frac {1}{3}}{\pi}{r^2}2 → 24={\frac {2}{3}}{r^2} → r^2=36 → r=±6$$
Since radius is a positive number, 𝑟 = 6.
Maps to Category 76 – Volume of spheres, cones, pyramids, and cylinders.

Step 1: Determine the end population of City X in 2015
Since it is given that the population of City X and City Y were same in 2015, this is also the end population of City Y in 2015.
20% increase in decimal from 2010 to 2015 = 1 + 0.2 = 1.2
Hence, the end population in 2015 was
120,000 × 1.2 = 144,000
Step 2: Determine the initial population of City Y in 2010
End population in 2015 = 144,000 from Step 1.
10% decrease in decimal from 2010 to 2015 = 1  0.1 = 0.9
Hence, the initial population in 2010 was
$$\frac {144000}{0.9}=160000$$
Maps to Category 46 – The original number before a percent increase/decrease.

Step 1: Rearrange the equation to isolate the radius
$$V={\frac {4}{3}}{\pi}{r^3} → {r^3}={\frac {3V}{4\pi}} → r=\sqrt[3]{\frac {3V}{4\pi}}$$
Maps to Category 41 – Rearranging variables in an equation.

Step 1: Determine the probability
$$probability ={\frac {desired\ \ outcomes}{total\ \ outcomes}}$$
Desired outcomes = Always = 30.9%.
Total outcomes = total of all answers except Never = 24.3% + 13.5% + 30.9% = 68.7%
$$probability ={\frac {30.9}{68.7}}=0.449=0.45$$ Maps to Category 52 – Probability.