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Category 15 Example 1 Question 1 - no calculation required

Total annual cost = annual membership + cost of p classes in a year = 25 + 20p


Category 15 Example 1 Question 2 - mental math

Cost of 5 classes is 20 x 5 = 100. Hence, total annual cost = 25 + 100 = 125


Category 15 Example 2 - mental math

Total books = existing books + 4 books per week for 7 weeks = 38 + 28 = 66


Category 16 Example 1 - no calculation required

Total earnings = h hours at $15 per hour + c cakes at $2.50 each = 15h + 2.5c


Category 17 Example 1 - no calculation required

The given equation can be interpreted as

total population in t years after 1998 = starting population in 1998 + average increase in population per year after 1998 for t years.

Hence, 130 is the average increase in population per year after 1998 for t years.


Category 17 Example 2 Question 1 - no calculation required

The given equation can be interpreted as

tusk size in m months after one year of birth = tusk size at one year + average increase in tusk size per month after one year of birth for m months.


Category 17 Example 2 Question 2 - no calculation required

For tusk size to increase by 1, 0.5583m must be equal to 1. Hence, m is the reciprocal of 0.5583 = 1/0.5583.


Category 18 Example 1 - no calculation required

Total bushes = x + y = 12

Total cost = cost of x bushes at $35 each + cost of y bushes at $55 each --> 520 = 35x + 55 y


Category 9 Example 1 - calculation not required

It can be observed that the values of a and b in the top equation are half of the bottom equation, but this is not true for the values of c. This implies that the ratios of a and b are same but different than that of c. Hence, the system has no solution.


Category 9 Example 2 - mental math for those good with dividing fractions

It can be observed that the values of a, b, and c in the top equation are six times of the bottom equation: 3 is six times of 1/2, 2 is six times of 1/3, and 6 is six times of 1. This implies that the ratios of a, b, and c are same. Hence, the system has infinitely many solutions.


Category 10 Example 1 - mental math

It can be observed that in the top equation the value of b is 4 times of the bottom equation. For the system to have no solution, the value of a in the top equation must also be 4 times of the bottom equation. (This will result in the same ratios of a and b.)

Since a = 1 in the bottom equation, it must be 4 in the top equation. Hence, 2n = 4 - -> n = 2.


Category 10 Example 2 - mental math

It can be observed that in the top equation the values of a and b are one-third of the bottom equation. For the system to have no solution, the value of c in the top equation cannot be one-third of the bottom equation. Hence k = c cannot be 1.


Category 11 Example 1 - mental math

It can be observed that in the top equation the value of b is 3 times of the bottom equation. Since the system has infinitely many solutions, the vales of a and c in the top equation must also be 3 times of the bottom equation (this will result in the same ratios of a, b, and c).

Hence, a = 12 and c = 3.

Category 11 Example 2 - mental math for those good with dividing fractions

It can be observed that in the top equation the value of c is 12 times of the bottom equation. Since the system has infinitely many solutions, the values of a and b in the top equation must also be 12 times of the bottom equation (this will result in the same ratios of a, b, and c).

Hence, a = 4 and b = 3.


Category 12 Example 1 - mental math

It can be observed that the value of b in the bottom equation is five times of the top equation. For the system to have one solution, the value of a = k in the bottom equation cannot be 5 times the value of a in the top equation. Hence, k cannot be 10.


Category 12 Example 3 - shortcut and may be mental math

The question asks for the value of 50x + 10y. It can be observed that if the two equations are added, the result is the value of 5x + y. The value can be multiplied by 10 to obtain the answer.


Category 14 Example 1 - mental math

Since the equation has infinitely many solutions, the coefficient of x in the left expression must be 9 and the constant in the right expression must be 6. Hence, k = 3 and c = 2.


Category 14 Example 4 - mental math

It can be observed that the constant in left and right expression is same. For the equation to have no solution, 0.5a cannot be 10. Hence, a cannot be 20.


Category 2 Example 1 - no calculation required

The slope and the y-coordinate of the y-intercept are given (a point with x = 0 is the y-intercept). Match them to the answer choices given as y = mx + b equation.


Category 2 Example 3 - no calculation required

The slope can be determined from the graph as rise/run. The y-intercept can be directly read from the graph. Match them to the answer choices given as y = mx + b equation.


Category 3 Example 1 - mental math

It can be read from the equation as C/B.


Category 3 Example 2 - mental math

The slope can be determined from the graph as rise/run. Match this with -A/B in each answer choice and eliminate the answer choices where the slope does not match.

The y-intercept can be read from the graph. Match it with C/B in the remaining answer choices.


Category 8 Example 1 - no calculation required

Use the transformation rule given in Key Points for left shift and match it with the answer choices.

Category 8 Example 2 - no calculation required

Match the reflected graph based on the transformation rule given in Key Points.

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