An artisan makes two types of handmade candles.
- Each standard candle requires pound of wax and burns for hours.
- Each deluxe candle requires pound of wax and burns for hours.
On a certain day, the artisan used a total of pounds of wax to make candles that will burn for a combined hours.
How many deluxe candles did the artisan make?
Two unknown candle counts and two separate totals point to a linear system: two equations about the same counts. Write one equation in pounds and another in hours, matching each per-candle rate to its candle type. Graph both equations in Desmos and click their intersection. Read the coordinate for the requested type: pounds of wax are not a count of candles.
Hints
- Hint 1
Let count standard candles and count deluxe candles. To find the wax total, multiply each type’s pounds per candle by its candle count, then add the two amounts. What should that sum equal?
- Hint 2
The burning-time equation must use hours throughout. Hours per candle multiplied by a candle count gives hours; wax amounts don’t belong in this equation. What two terms add up to the combined burning time?
- Hint 3
An intersection is a point on both graphs, so its coordinates satisfy both totals. After clicking it in Desmos, read the second coordinate: you defined as the number of deluxe candles.
Step-by-step
Approach 1: Graph the two totals in Desmos
Step 1Write the wax equation
Let be the number of standard candles and the number of deluxe candles. Multiply each wax amount per candle by its count, then add to get the pounds used: . Keep each rate with its own candle type.
- Step 2
Write the burning-time equation
Each standard candle contributes hours, and each deluxe candle contributes hours. Their hours, not their pounds of wax, add to the given total:
- Step 3
Read the deluxe count at the intersection
Type both equations on separate Desmos lines and click their intersection, the point that satisfies both equations. Desmos shows . The first coordinate is , the standard count; the second is , so the artisan made deluxe candles. Choice C.
Approach 2: Cancel the standard-candle count
Step 1Scale the wax equation to match standard hours
To match the in the hours equation, multiply the wax equation by , since :
- Step 2
Subtract to remove standard candles
Subtract that scaled equation from the burning-time equation. The standard-candle terms cancel, leaving only deluxe candles:
- Step 3
Find the deluxe count
Divide by to find , the number of deluxe candles:
. The artisan made deluxe candles. Choice C.