**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**

**Question 21 Answer B**

Step 1: Plug
in the given value of π¦ in the equation and solve for π₯

π₯ = 10%
saline.

π¦ = 20% saline = 100 ml.

0.1π₯ + (0.2 × 100) = 0.18(π₯ + 100) → 0.1π₯ + 20 = 0.18π₯ + 18 → 0.08π₯ = 2 → π₯ = 25

There are several categories that
demonstrate how to plug values in an equation and solve.

**Question 22 Answer D**

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.

**Question 23 Answer A**

Step 1:
Determine the slope

The question
mentions that the relationship between π₯ and π¦ is linear. This implies that
each row in the table is a point (π₯, π¦) on a line. Select any two points from the table and
determine the slope. Points (π, 0) and (3π, –π) are used below.

$${\frac {-a-(-0)}{3a-a}} = {\frac {-a}{2a}} =-{\frac {1}{2}}$$

Step 2:
Determine the π¦-intercept

Plug in the slope and any (π₯, π¦) point in π¦ = ππ₯ + π equation and determine π (π¦-intercept). Point (π, 0) is used below.

$$0=-{\frac {1}{2}}a+b → b={\frac {a}{2}}$$

The answer choices are given in the standard form equation, ππ₯ + ππ¦ = π. If you are familiar with matching the slope and the π¦-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 π¦-intercept in π¦ = ππ₯ + π 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.

**Question 24 Answer B**

Step 1: Determine the slope of line of best fit

The equation for the line of best fit is the linear
equation π¦ = ππ₯ + π.

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.

**Question 25 Answer A**

Step 1: Solve for π₯

When the equation 1.6π₯ + 0.5π¦ = –1.3 is multiplied by 3, both equations will have 1.5π¦ that can be canceled out, as shown below.

2.4π₯ – ~~1.5π¦~~ = 0.3

4.8π₯ + ~~1.5π¦~~ = – 3.9

-------------------------

7.2π₯ = – 3.6

π₯ = – 0.5

Maps to Category 11 – System of linear equations with one solution.

**Question 26 Answer D**

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

$$y=a(x)^t$$

π = initial value = 310.

π₯ = rate of change as percent decimal = 1 + 0.01 = 1.01.

π‘ = time.

π¦ = end value in π‘ years = π(π‘).

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.

**Question 27 Answer A**

Step 1: Solve

(9π₯ – 6)
can be factored as 3(3π₯ – 2). The equation becomes

$${\frac {2}{3}}{\times 3}(3x-2)-4=3(3x-2) →$$

$$2(3x-2)-4=3(3x-2)$$

Move 3π₯ – 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.

**Question 28 Answer D**

Step 1: Determine
whether the parabola opens upwards or downwards

The
equation of the parabola is given in the factored form π¦ = π(π₯ + π)(π₯ + π) where π and π are the π₯-intercepts of the parabola.

Since π is positive, the parabola opens upwards. This eliminates answer choices A and B.

Step 2: Match
the π₯-intercept in the given equation with the graph

From the
equation the π₯-intercepts are –3 and π. This eliminates answer choice C that
does not have an π₯-intercept = –3.

Maps to Category 32
– Factored form equation of a parabola.

**Question 29 Answer D**

Step 1: Determine
the components of the linear equation

The
height is the sum of the constant 32.01 and the unknown number 1.88πΏ.

The value given
to πΏ in the unknown number 1.88πΏ determines the increase in the height.

For every 1
inch of πΏ (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.

**Question 30 Answer A**

Step 1: Determine an approach

A student must
know the special right triangle 30-60-90 to solve this question.

Let πΆπ· = π.
Since πΆπ· is half of π΄π΅, π΄π΅ = 2π.

See figure below. A perpendicular
line from point B creates the figure π΅πΆπ·πΈ with all corners as 90 degrees angles
and the triangle π΄π΅πΈ.

Step 2: Determine angle π΄π΅πΈ

Since π΅πΈ and πΆπ· are between
the parallel lines π΅πΆ and π΄π·, the lengths of π΅πΈ and πΆπ· are same = π.

Hence, in triangle π΄π΅πΈ, π΅πΈ = π and π΄π΅ = 2π. 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 π΄π΅πΈ is opposite side to the side π΄πΈ, angle π΄π΅πΈ = 60 degrees.

Step 3: Determine angle π΅

angle π΅ = 90 + 60 = 150

Maps to Category 71 – Angles, sides, and area of a triangle.

**Question 31 Answer 6**

Step 1: Determine
the conditional relationship

Let the number
of oranges = π₯.

The total cost
of 5 apples and π₯ oranges should be less than or equal to 8. As an inequality, this can be written as

cost of
5 apples + cost of π₯ oranges ≤ 8

Cost of 5
apples = 5 × 0.65 = 3.25.

Cost of π₯ oranges = 0.75π₯.

Hence,

3.25 + 0.75π₯ ≤ 8

Step 2: Solve
for the maximum number of oranges

0.75π₯ ≤ 8 – 3.25 → 0.75π₯ ≤ 4.75 → π₯ ≤ 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.

**Question 32 Answer 146**

Step 1: Determine the
value of π + π

The sum of the
angles of a triangle is 180 degrees. Hence,

π + π + π = 180

Plug in the given value of π.

34 + π + π = 180 → π + π = 180 – 34 = 146

Maps to Category 71
– Angles, sides, and area of a triangle.

**Question 33 Answer 2500**

Step 1: Set up the mean and solve for π₯

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, π₯.

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.

**Question 34 Answer 34**

Step 1: Determine 2π

The
given equation is the slope intercept equation of a line where π = 0. Since π = 0, the change in π₯ and π¦ are proportional to each other for any value of the constant π (slope). Doubling
the value of π₯ will double the value of π¦.

For π₯ = π, π¦ = 17. Hence, for π₯ = 2π

π¦ = 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.

ππ₯ + ππ = 4π₯ + 10

Equate the coefficients.

π = 4

Equate the constants.

ππ = 10

Step 2: Determine π

Plug in π = 4 in ππ.

$${4\times b}=10 → b={\frac {10}{4}}={\frac {5}{2}}=2.5$$

Maps to Category 13
– Solutions of linear expressions.

**Question 36 Answer 6.25**

Step 1: Determine the point of intersection

π¦ = π 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 π and the π¦-coordinate of the vertex are same.

Step 2: Determine the π₯-coordinate of the parabola vertex

In the given equation, π = –1 and π = 5.

$$x\ coordinate =-{\frac {b}{2a}}=-{\frac {5}{2\times -1}}={\frac {5}{2}}=2.5$$

Step 3: Determine the π¦-coordinate of the parabola vertex

Plug π₯ = 2.5 in the given equation to determine π¦ = π.

$$y=-(2.5)^2+(5\times 2.5)=-6.25+12.5=6.25$$

Maps to Category 34 – Parabola intersections and system of equations.

**Question 37 Answer 293**

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.

**Question 38 Answer 9**

Step 1: Set up a proportion for miles and seconds

The speed is given as 200 miles per hour. This is same as 200 miles in 3600 seconds.

Falcon dive distance = 0.5 miles. Let seconds to dive 0.5 miles = π₯.

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

Maps to Category 44 – Rate.