Heart of Algebra Previous Year Questions — SAT Math
17 past-year Heart of Algebra questions from SAT (Math) — page 3.
Q10 (2023)
In a factory, the number of units manufactured in a month follows the linear function $N = 231D$, where $N$ is the number of units produced in the month and $D$ is the number of days elapsed in the month. Which of the following represents 231 in the above equation?
- The number of units manufactured only in the first day
- The number of workers working in the factory
- The number of units manufactured each day of the month
- The increase in the number of workers in the factory per day
### Related Formula
$N = mD$
where $m$ represents the rate of unit production per day.
### Core Logic
Evaluate $N$ for successive values of $D$:
For $D = 1$, $N = 231$.
For $D = 2$, $N = 462$.
Each elapsed day adds 231 units to total production.
### Step 1: Conclusion
Thus, 231 represents the number of units manufactured each day of the month (Option C).
### Pattern Recognition
Linear multiplier of time variable $\rightarrow$ represents daily rate.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q12 (2023)
If $5a - 7 = 2a - 22$, what is the value of $a + 2$?
- 4
- -3
- 0
- 2
### Related Formula
$$ax + b = cx + d \implies (a - c)x = d - b$$
### Core Logic
Solve $5a - 7 = 2a - 22$ for $a$:
$$5a - 2a = -22 + 7$$
$$3a = -15 \implies a = -5$$
### Step 1: Calculate Target Expression
Substitute $a = -5$ into $a + 2$:
$$a + 2 = -5 + 2 = -3$$
Thus, option B is correct.
### Pattern Recognition
Trap alert: do not stop at $a = -5$; always check what the question asks for ($a + 2$).
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q13 (2023)
$P = 89 + 12t$
The number of vehicles in public transportation in City X between 2005 and 2014 is modeled by the equation above. In the equation, $P$ is the number of vehicles and $t$ is the year (assuming 2005 is $t = 0$). What does the 12 mean in the equation?
- The number of vehicles in public transportation was 12 in 2005.
- The number of vehicles in public transportation increased by 12 vehicles each year between 2005 and 2014.
- The number of vehicles in public transportation increased in total by 12 between 2005 and 2014.
- The number of vehicles in public transportation was 12 in 2014.
### Related Formula
$y = mt + b$
where $m$ represents the constant rate of change per unit of time $t$.
### Core Logic
In $P = 89 + 12t$, $89$ is the initial value at $t = 0$ (year 2005), and $12$ is the rate of change per year.
Each increase of $1$ in $t$ causes $P$ to increase by $12$.
### Step 1: Conclusion
Thus, 12 means the number of vehicles increased by 12 vehicles each year between 2005 and 2014 (Option B).
### Pattern Recognition
Coefficient of time variable $t$ $\rightarrow$ represents annual rate of increase/decrease.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q14 (2023)
$$f(x) = x^2 + px + q$$
If $p = 2q$ and the function has a value of 16 at $x = 3$, what is the value of $p$?
### Related Formula
$$f(3) = (3)^2 + p(3) + q = 16$$
Given substitution relationship $p = 2q$.
### Core Logic
Substitute $p = 2q$ into $f(x)$:
$$f(x) = x^2 + 2qx + q$$
Substitute $x = 3$ and $f(3) = 16$:
$$16 = 3^2 + 2q(3) + q$$
$16 = 9 + 6q + q$
$16 = 9 + 7q$
$$7q = 7 \implies q = 1$$
### Step 1: Calculate p
Calculate $p$ using $p = 2q$:
$p = 2(1) = 2$
Thus, the answer is 2.
### Pattern Recognition
Substitute known relation $p = 2q$ to convert function into single parameter equation.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q15 (2023)
In the TAS aptitude test, three marks are awarded for a correct response and one mark is deducted for an incorrect response. John took the test and secured 76 marks. If there are 40 questions in the test and John attempted all questions, how many of his responses were incorrect?
### Related Formula
$c + i = N$
$3c - 1i = S$
where $c$ is correct responses, $i$ is incorrect responses, $N$ is total questions, and $S$ is score.
### Core Logic
Set up system of equations:
$$c + i = 40 \quad \dots (1)$$
$$3c - i = 76 \quad \dots (2)$$
Add (1) and (2) to eliminate $i$:
$$4c = 116 \implies c = 29$$
### Step 1: Solve for Incorrect Responses
Substitute $c = 29$ into equation (1):
$$29 + i = 40 \implies i = 40 - 29 = 11$$
John had 11 incorrect responses.
### Pattern Recognition
Test marking system $\rightarrow$ correct + incorrect = total questions; 3(correct) - 1(incorrect) = score.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
