A truck delivered n number of boxes weighing 20 pounds and t number of boxes weighing 30 pounds to a warehouse. The equation w = 20n + 30t can be used to determine the total weight w, in pounds, of the boxes. If the total weight of the boxes delivered was 480 pounds and 12 boxes weighed 30 pounds, how many boxes weighed 20 pounds?


Maps to Category 15 of the book.

Answer posted on 08/01/2021

Updated: 3 days ago

Below are solutions to questions that can be solved without calculation or can be solved using mental math after completing the book.

Detailed solutions to all the questions on Section 3 and 4 of this test are available at https://www.tutorhubllc.com/math-corner.

You will need the test to follow through. It is available at https://www.tutorhubllc.com/sat-practice-tests.

Section 3 Question 1 - mental math

Collect variables on one side and numbers on the other side.

This gives 2z - z = -1 --> z = -1.

Correct answer = B.


Section 3 Question 2 - no calculation required

Total purchase price = initial payment + $60 per week for w weeks -->

300 = 30 + 60w

Correct answer = C.


Section 3 Question 3 - mental math, depending on comfort level

Since the relationship between weight and shipping charge is linear, each row in the table is a (x, y) point on a linear line. x = weight and y = shipping charge.

The equations in the answer choices are in the slope-intercept form y = mx + b. The y-intercept = b is the shipping charge y when weight x = 0.


Without calculation, it can be observed from the first two rows that for every 5 pounds of weight the shipping charge is approximately $5 (approximately 22 - approximately 17 = 5).

Hence, for 0 pounds of weight the shipping charge is approximately 17-5 =12. This eliminates answer choices A and C that have y-intercept = 0 and answer choice D that has y-intercept = 16.94.

Correct answer = B.


Section 3 Question 4 - mental math

From the graph read y = height for each given x = base diameter.

It can be observed from the graph that for x = 5, y = 35 and for x = 2, y =14. The difference in the two heights y is 35 -14 = 21.

Correct answer = C.


Section 3 Question 5 - mental math

The square root of 9 is plus and minus 3. The square root of x squared is plus and minus x. Since x > 0, ignore the negative values. Hence, the square root is 3x.

Correct answer = A.


Section 3 Question 6 - mental math, depending on comfort level

x squared - 1 has two factors: (x - 1) and (x + 1). (x -1) from the numerator and the denominator cancel out simplifying the fraction to x + 1 = -2. Hence, x = -3.

Correct answer = A.


Section 3 Question 7 - no calculation required

f(0) is the value of y for x = 0.

From the graph, it can be observed that for x = 0, the value of y is 4.

Correct answer = D.


Section 3 Question 8 - mental math, depending on comfort level

Angle CBD is a complimentary angle = 90 degrees. Hence, all angles constituting angle CBD add to 90.

x + 2x + 2x = 90 --> 5x = 90 --> x = 18.

Use approximation to determine 3x. 3x must be 54 since numbers in other answer choices are either smaller or larger to be correct.

Correct answer = C.


Section 3 Question 9 - mental math

Since the line is slanting downwards from left to right, the slope is negative. Match answer choices for negative slope. This eliminates answer choices A and B (for negative slope, values of both a and b must be positive or must be negative).

From the graph the y-intercept = -4. This eliminates answer choice D.

Correct answer = C.


Section 3 Question 10 - no calculation required

The equation of the parabola is given in the standard form.

It can be observed from the graph that k is the y-coordinate of the y-intercept of the parabola.

In the standard form equation, this is the value of c. Hence, k = c = 12.

Correct answer = D.


Section 3 Question 11 - mental math

Match the given coordinates of the center and the square of the given radius with the answer choices. Answer choice A matches.

Correct answer = A.


Section 3 Question 12 - mental math

Since the two triangles are similar, cos E = cos B. The length of all 3 sides of triangle ABC are given. Hence, cos B = adjacent/hypotenuse = 12/13.

Correct answer = B.


Section 3 Question 13 - mental math, depending on comfort level

The two x-intercepts are the two values of x in the given equation.

The equation can be factored without the quadratic formula. The two values of x that add up to b = 5 and are multiples of c = 4 are 4 and 1. Hence the two factors are (x + 4) and (x + 1) and the two values of x are -4 and -1.

The distance between -4 and -1 is 3.

Correct answer = C.


Section 3 Question 15 - mental math

It can be observed that the value of b in the bottom equation is twice of the top equation.

For the system to have no solution, the value of a in the bottom equation must also be twice of the top equation. Hence, a must be 2 times of -3 = -6.

Correct answer = A.


Section 3 Question 19 - mental math

The question asks for the value of 5x + 5y.

It can be observed that if the x and the y variables are added in both the equations the result will be 5x + 5y = 2500.

Correct answer = 2500.


Section 3 Question 20 - mental math, depending on comfort level

(u squared - t squared) has two factors (u - t) and (u + t). The equation becomes (u - t)(u - t)(u + t). Substitute the given values: 2 x 2 x 5 = 20.

Correct answer = 20.


Section 4 Question 1 - no calculation required

Total height of helicopter = initial height above ground + 21 feet per second for t seconds -->

y = 40 + 21t

Correct answer = B.


Section 4 Question 2 - no calculation required

The price does not change for first 100 texts. After 100 texts (x-axis), the price (y-axis) increases.

Hence, the graph will not show an increase in cost = y till texts = x = 100. After that, cost y will increase with number of texts x. This matches the graph in answer choice A.

Correct answer = A.


Section 4 Question 3 - no calculation required

Since the popcorns are being eaten, they can not increase at any time point. This eliminates answer choices A, C, and D that show an increase at some time point (increase with time is represented by a slanting upwards line).

Correct answer = B.


Section 4 Question 4 - mental math

Collect variables on one side and numbers on the other side.

Moving x to the right side and 15 to the left side gives 20 - 15 = x --> x = 5. 3x will be 15.

Correct answer = C.


Section 4 Question 5 - mental math

Substituting x = -1 in the given equation gives 2/2 = 1.

Correct answer = C.


Section 4 Question 6 - mental math

Simplify the parentheses. (Multiply each term within the parentheses with 2x.)

Correct answer = D.


Section 4 Question 7 - no calculation required

The sample must be random and represent the sample population. This eliminates answer choices A and C since the sample is not random. Answer choice D does not represent a true survey.

Correct answer = B.


Section 4 Question 10 - mental math

Margin of error is the plausible negative and positive deviation by which the results of a survey or a study are expected to differ. Hence, margin of error is between 23 - 4 = 19 and 23 + 4 = 27.

Correct answer = D.


Section 4 Question 13 - no calculation required

It can be observed from bar graph that colony C increased after 2 weeks whereas Colony A and Colony B decreased every 2 weeks. hence option II and III are correct.

Correct answer = C.


Section 4 Question 19 - no calculation required

Since a is negative, the vertex is at maximum. From the equation the coordinates of vertex are (3, a).

Correct answer = D.


Section 4 Question 20 - mental math

In the original data set 84 is the highest number. The new data set has one extra number 96 that is 12 higher than 84. It will increase the range of the numbers by 12.

Since the actual numbers in the data set are not given, it is not be determined if the other measures increased by 12 or not.

Correct answer = C.


Section 4 Question 22 - no calculation required

Since the number of people invited doubled each year, the relationship between the number of people invited each year and time in years is exponential increase.

Correct answer = D.


Section 4 Question 26 - mental math

All the answer choices are given as exponential function.

The number of pollen grains per square centimeter in first year = 310 = initial value.

Each year this increased by 1% = 0.1. Hence, rate of change per year is 1 + 0.1 = 1.01. This matches answer choice D

Correct answer =D .


Section 4 Question 28 - no calculation required

Since a is positive, answer choices A and B where parabola opens downwards can be eliminated.

From the given equation, the two x-intercepts are -3 and k. This eliminates answer choice C since the parabola does not have an x-intercept = -3.

Correct answer = D.


Section 4 Question 29 - no calculation required

When L = 1, H will increase by 1.88 (1.88 x 1= 1.88). Hence, 1.88 is the increase in H = height when L = length of femur = 1

Correct answer = D.









Updated: 5 days ago


Category 80 Example 1 Question 1 - mental math

Since cos A is 5/13, 13 is the hypotenuse and 5 is side adjacent to angle A. Since 5 and 13 are the ratio of Pythagorean triple 5:12:13, third side opposite to angle A must be 12. Hence, sin A = opposite/hypotenuse = 12/13.


Category 80 Example 1 Question 2 - no calculation required

cosine of an angle equals the sine of its complement.


Category 80 Example 2 Question 1 - mental math

Since the two triangles are similar, sin F = sin C. It is given that sin A is 0.8= 8/10. Hence, 10 is the hypotenuse and 8 is the side opposite to angle A. Since 8 and 10 are the ratio of Pythagorean triple 6:8:10, third side adjacent to angle A but opposite to angle C must be 6. Hence sin C = opposite/hypothenuse = 6/10 = 3/5 = sin F.