Question 99·Hard·Linear Equations in One Variable
In a fundraising campaign, each small candle sold brings in a profit of $1.20, and each large candle sold brings in a profit of $2.80. The campaign must raise exactly $640 in profit and plans to sell exactly 400 candles in total.
If is the number of small candles the campaign sells, which equation can be used to determine ?
For word problems asking you to write an equation, always start by clearly defining the variable exactly as the problem states, then express every other quantity in terms of that variable (for example, using a total like 400 to write ). Next, convert each rate statement (like $1.20 per small candle) into a multiplication expression, add or subtract these expressions as the story describes, and finally set the result equal to the given total (such as 640). Once you have your equation, compare its structure—term by term—to each answer choice instead of solving it, which is usually faster on the SAT. Соntent by Aniko.aі
Hints
Relate the two candle types
If is the number of small candles and there are 400 candles total, how can you write the number of large candles in terms of ?
Turn profits into algebraic expressions
What expression represents the profit from small candles if each small candle gives a profit of and there are of them? What about the profit from large candles using the number you found in the first hint?
Use the total profit condition
Once you have expressions for the profit from small and large candles, how can you combine them and use the fact that the total profit must be ?
Desmos Guide
Represent each answer choice as a function
In Desmos, enter the left-hand side of each answer choice as a separate function, for example:
fA(x) = 1.20x + 2.80xfB(x) = 1.20x + 2.80(640 - x)fC(x) = 1.20(400 - x) + 2.80xfD(x) = 1.20x + 2.80(400 - x)These functions give the total profit (in dollars) as a function of the number of small candles (or as each equation defines it).
Use the required total profit as a horizontal line
Add the line y = 640 to represent the required total profit. For each function, look for the point(s) where its graph intersects this horizontal line; those -values are where that equation says the total profit is 640.
Check which model matches both story conditions
Among the functions, focus on the one that: (1) uses specifically for small-candle profit, (2) uses multiplied by the number of large candles written as "total candles minus ", and (3) equals 640 for the total profit (the right-hand side of the equation). The equation in the choices that has this structure is the one you should select. аnikо.aі
Step-by-step Explanation
Translate the candle counts into algebra
The problem defines as the number of small candles.
Since there are 400 candles in total, the number of large candles must be the total minus the small ones:
So we now have:
- Small candles:
- Large candles: .
Write expressions for profit from each candle type
Each small candle brings in a profit of , and there are small candles. So the total profit from small candles is:
Each large candle brings in a profit of , and there are large candles. So the total profit from large candles is:
The total profit is the sum of these two expressions. aniкo.аi/ѕat
Set up the total-profit equation and match it to a choice
The campaign must raise exactly in profit. That means:
(profit from small candles) (profit from large candles) (total profit)
Using the expressions from Step 2, we get:
Now compare this with the answer options: it matches choice D) , so D is correct.