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### SAT Math Test 10 Section 4 solved questions 21 to 38

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 21 to 38 Answer Explanations

Step 1: Plug in the given value of 𝑦 in the equation and solve for 𝑥
𝑥 = 10% saline.
𝑦 = 20% saline = 100 ml.
0.1𝑥 + (0.2 × 100) = 0.18(𝑥 + 100)  →  0.1𝑥 + 20 = 0.18𝑥 + 18    0.08𝑥 = 2  →  𝑥  = 25
There are several categories that demonstrate how to plug values in an equation and solve.

Step 1: Determine the relationship between time and the number of people
Since the number of number of people invited each year doubled than the previous year, the function is exponential increase.
Maps to Category 50 – Linear versus exponential growth and decay.

Step 1: Determine the slope
The question mentions that the relationship between 𝑥 and 𝑦 is linear. This implies that each row in the table is a point (𝑥, 𝑦) on a line. Select any two points from the table and determine the slope. Points (𝑎, 0) and (3𝑎, –𝑎) are used below.
$${\frac {-a-(-0)}{3a-a}} = {\frac {-a}{2a}} =-{\frac {1}{2}}$$
Step 2: Determine the 𝑦-intercept
Plug in the slope and any (𝑥, 𝑦) point in 𝑦 = 𝑚𝑥 + 𝑏 equation and determine 𝑏 (𝑦-intercept). Point (𝑎, 0) is used below.
$$0=-{\frac {1}{2}}a+b → b={\frac {a}{2}}$$
The answer choices are given in the standard form equation, 𝑎𝑥 + 𝑏𝑦 = 𝑐. If you are familiar with matching the slope and the 𝑦-intercept in a standard form equation, then you know the correct answer is A. If not, proceed to step 3.
Step 3: Determine the standard form equation
Plug in the slope and the 𝑦-intercept in 𝑦 = 𝑚𝑥 + 𝑏 equation and rearrange the equation to the standard form.
$$y=-{\frac {1}{2}}x+{\frac{a}{2}} → 2y=-x+a → x+2y=a$$
Maps to Category 1 – Line equation in slope intercept form and Category 2 – Line equation in standard form.

Step 1: Determine the slope of line of best fit
The equation for the line of best fit is the linear equation 𝑦 = 𝑚𝑥 + 𝑏
If you are good at slope, it can be seen from the graph that slope = rise/run = 0.5. Only answer choice B matches the slope.
Alternatively, select any two points on the line to determine the slope using the slope equation. Points (140, 70) and (200, 100) are selected for the calculation below.
$$\frac {100-70}{200-140}=\frac {30}{60}=0.5$$
This eliminates answer choices A, C, and D.
Maps to Category 54 – Scatter plots and line of best fit.

Step 1: Solve for 𝑥
When the equation 1.6𝑥 + 0.5𝑦  = –1.3 is multiplied by 3, both equations will have 1.5𝑦 that can be canceled out, as shown below.
2.4𝑥  1.5𝑦  = 0.3
4.8𝑥 + 1.5𝑦  = – 3.9
-------------------------
7.2𝑥 = – 3.6
𝑥 = – 0.5
Maps to Category 11 – System of linear equations with one solution.

Step 1: Determine the components of the exponential growth equation
Since each year the grains deposited increased by 1% than the grains deposited the previous year, the question is on exponential growth. The equation for exponential growth is
$$y=a(x)^t$$
𝑎 = initial value = 310.
𝑥 = rate of change as percent decimal = 1 + 0.01 = 1.01.
𝑡 = time.
𝑦 = end value in 𝑡 years 𝑃(𝑡)
Step 2: Determine the equation
Plug the above values in the equation.
$$y=a(x)^t → P(t)=310(1.01)^t$$
Maps to Category 51 – Exponential growth and decay.

Step 1: Solve
(9𝑥  6) can be factored as 3(3𝑥 – 2). The equation becomes
$${\frac {2}{3}}{\times 3}(3x-2)-4=3(3x-2) →$$
$$2(3x-2)-4=3(3x-2)$$
Move 3𝑥 – 2 terms to one side.
$$3(3x-2)-2(3x-2)=-4 → (3x-2)=-4$$
There are several categories that demonstrate how to solve equations.

Step 1: Determine whether the parabola opens upwards or downwards
The equation of the parabola is given in the factored form 𝑦 = 𝑎(𝑥 + 𝑚)(𝑥 + 𝑛) where 𝑚 and 𝑛 are the 𝑥-intercepts of the parabola.
Since 𝑎 is positive, the parabola opens upwards. This eliminates answer choices A and B.
Step 2: Match the 𝑥-intercept in the given equation with the graph
From the equation the 𝑥-intercepts are –3 and 𝑘. This eliminates answer choice C that does not have an 𝑥-intercept = –3.
Maps to Category 32 – Factored form equation of a parabola.

Step 1: Determine the components of the linear equation
The height is the sum of the constant 32.01 and the unknown number 1.88𝐿.
The value given to 𝐿 in the unknown number 1.88𝐿 determines the increase in the height.
For every 1 inch of 𝐿 (length of the femur), the height will increase by 1.88.
32.01+ (1.88 × 1) = 32.01 + 1.88
Maps to Category 16 – Word problems on interpretation of linear equations.

Step 1: Determine an approach
A student must know the special right triangle 30-60-90 to solve this question.
Let 𝐶𝐷 = 𝑎. Since 𝐶𝐷 is half of 𝐴𝐵, 𝐴𝐵 = 2𝑎.
See figure below. A perpendicular line from point B creates the figure 𝐵𝐶𝐷𝐸 with all corners as 90 degrees angles and the triangle 𝐴𝐵𝐸.
Step 2: Determine angle 𝐴𝐵𝐸
Since 𝐵𝐸 and 𝐶𝐷 are between the parallel lines 𝐵𝐶 and 𝐴𝐷, the lengths of 𝐵𝐸 and 𝐶𝐷 are same = 𝑎.
Hence, in triangle 𝐴𝐵𝐸, 𝐵𝐸 = 𝑎 and 𝐴𝐵 = 2𝑎. These are the side ratios of a 30-60-90 triangle. The ratio of the sides in relation to opposite angles is
$$30^{\circ}:60^{\circ}:90^{\circ} → {a:}\ a{\sqrt{3}}\ :2a → BE: AE:AB$$
Since angle 𝐴𝐵𝐸 is opposite side to the side 𝐴𝐸, angle 𝐴𝐵𝐸 = 60 degrees.
Step 3: Determine angle 𝐵
angle 𝐵 = 90 + 60 = 150
Maps to Category 71 – Angles, sides, and area of a triangle.

Step 1: Determine the conditional relationship
Let the number of oranges = 𝑥.
The total cost of 5 apples and 𝑥 oranges should be less than or equal to 8. As an inequality, this can be written as
cost of 5 apples + cost of 𝑥 oranges ≤ 8
Cost of 5 apples = 5 × 0.65 = 3.25.
Cost of 𝑥 oranges = 0.75𝑥.
Hence,
3.25 + 0.75𝑥 ≤ 8
Step 2: Solve for the maximum number of oranges
0.75𝑥 ≤ 8 – 3.25  →  0.75𝑥 ≤ 4.75  →  𝑥 ≤ 6.33
Hence, number of oranges cannot be equal to or greater than 6.33. Since oranges are a whole number, the maximum number of oranges is 6.
Maps to Category 18 – Word problems on linear inequalities.

Step 1: Determine the value of 𝑏 + 𝑐
The sum of the angles of a triangle is 180 degrees. Hence,
𝑎 + 𝑏 + 𝑐 = 180
Plug in the given value of 𝑎.
34 + 𝑏 + 𝑐 = 180  →  𝑏 + 𝑐 = 180  34 = 146
Maps to Category 71 – Angles, sides, and area of a triangle.

Step 1: Set up the mean and solve for 𝑥
Mean is the sum of numbers in a data set divided by the count of numbers in the data set.
There are 5 numbers in the given data set: 700, 1200, 1600, 2000, 𝑥.
The mean of the data set is given as 1600.
Set up the mean and equate it to 1600.
$$mean=1600=\frac {700+1200+1600+2000+x}{5}$$
$$1600\times 5=5500+x → 8000=5500+x → x=2500$$  Maps to Category 57 – Mean.

Step 1: Determine 2𝑎
The given equation is the slope intercept equation of a line where 𝑏 = 0. Since 𝑏 = 0, the change in 𝑥 and 𝑦 are proportional to each other for any value of the constant 𝑚 (slope). Doubling the value of 𝑥 will double the value of 𝑦.
For 𝑥 = 𝑎, 𝑦 = 17. Hence, for 𝑥 = 2𝑎
𝑦 = 2 × 17 = 34
Maps to Category 1 –  Line equation in slope intercept form.

Question 35 Answer 5/2 or 2.5
Step 1: Equate the coefficients and the constants
For the equation to have infinitely many solutions, the expressions on both side of the equation must be equal.
Open parenthesis in the left side expression before equating.
𝑎𝑥 + 𝑎𝑏 = 4𝑥 + 10
Equate the coefficients.
𝑎 = 4
Equate the constants.
𝑎𝑏 = 10
Step 2: Determine 𝑏
Plug in 𝑎 = 4 in 𝑎𝑏.
$${4\times b}=10 → b={\frac {10}{4}}={\frac {5}{2}}=2.5$$
Maps to Category 13 – Solutions of linear expressions.

Step 1: Determine the point of intersection
𝑦 𝑐 is an equation of a horizontal line. Since it intersects the parabola at one point, it is a horizontal line tangent to the vertex of the parabola. This means that 𝑐 and the 𝑦-coordinate of the vertex are same.
Step 2: Determine the 𝑥-coordinate of the parabola vertex
In the given equation, 𝑎 =  1 and 𝑏 = 5.
$$x\ coordinate =-{\frac {b}{2a}}=-{\frac {5}{2\times -1}}={\frac {5}{2}}=2.5$$
Step 3: Determine the 𝑦-coordinate of the parabola vertex
Plug 𝑥 = 2.5 in the given equation to determine 𝑦 𝑐.
$$y=-(2.5)^2+(5\times 2.5)=-6.25+12.5=6.25$$
Maps to Category 34 – Parabola intersections and system of equations.

Step 1: Convert miles per hour to feet per hour
It is given that 1 mile = 5280 feet. Hence,
200 miles per hour = 200 × 5280 feet per hour
Step 2: Convert feet per hour to feet per second
Since 1 hour = 3,600 seconds, divide the above results by 3,600.
$${\frac {200\times 5280}{3600}} = 293.33=293\ feet\ per\ second$$
An alternate approach is to step up conversion factors if you are good at them.
Maps to Category 44 – Rate.

$${\frac {miles} {seconds}}={\frac{200}{3600}}={\frac {0.5}{x}} → x=9$$