Mathematics (Syllabus D) 4024/21 — May/June 2017

Walkthrough

We are given the total number of toy vehicles and the specific counts for cars and vans. To find the count for trucks, we subtract the known counts from the total. This gives us three numbers representing the quantities of each vehicle type. A ratio compares these quantities side-by-side. To put a ratio in its simplest form, we divide all parts by their greatest common divisor (or Highest Common Factor). In this case, dividing 24, 12, and 18 by 6 yields the simplest integer ratio.

Key Takeaways

• When given a total and some parts, the remaining part is found by subtraction: $\text{Part}_3 = \text{Total} - (\text{Part}_1 + \text{Part}_2)$.
• A ratio should always be expressed in its simplest whole-number form by dividing all terms by their HCF.

Common Mistakes

• Forgetting to calculate the number of trucks and using the total (54) or just one of the other numbers as the third term in the ratio.
• Simplifying the ratio incorrectly (e.g., only dividing two of the terms).
• Writing the ratio in the wrong order (e.g., Trucks : Cars : Vans instead of Cars : Vans : Trucks).

Things to Be Careful About

• Ensure you identify which quantities correspond to 'cars', 'vans', and 'trucks' before writing the ratio, as the question specifies the order "cars to vans to trucks".

Walkthrough

The key to this problem is introducing a scaling factor (often called $k$ or $x$) to represent the units in the new ratio $2:2:1$. Since the remaining vehicles are in the ratio $2:2:1$, we can say the remaining cars are $2k$, remaining vans are $2k$, and remaining trucks are $1k$. We then relate these back to the original counts by subtracting the sold amounts ($c, v, t$). This gives us three equations linking the sold amounts to $k$.

Next, we use the constraint "at least one of each type is sold" to find the range of possible integer values for $k$. Specifically, the number of vans sold ($v$) provides the tightest constraint because there are fewer vans initially than cars or trucks. Once we have the possible values for $k$, we look at the expression for the total number of vehicles sold. The total sold is the original total minus the remaining total ($5k$). To sell the smallest number of vehicles, we must keep the largest number of vehicles. Therefore, we choose the largest possible valid $k$.

Key Takeaways

• Ratios can be converted into algebraic expressions by multiplying each part by a constant $k$.
• Constraints like "at least one sold" translate into inequalities for the variables.
• Minimizing a sum of variables related linearly to a parameter often involves maximizing or minimizing that parameter within its valid range.

Common Mistakes

• Assuming the ratio $2:2:1$ applies to the sold vehicles instead of the remaining ones.
• Failing to check that the calculated sold amounts ($c, v, t$) are positive integers.
• Choosing the smallest $k$ (e.g., $k=1$) instead of the largest, leading to the maximum rather than minimum number of vehicles sold.
• Not verifying that the resulting remaining counts actually form the $2:2:1$ ratio.

Things to Be Careful About

• Remember that $k$ must be an integer because you cannot have a fraction of a vehicle.
• Double-check the direction of the inequality when solving for $k$ from the condition $v \geq 1$. Since $v = 12 - 2k$, increasing $k$ decreases $v$, so there is an upper bound on $k$.
• The question asks for $c, v, t$ specifically, not just the total sold.