After a manufacturer reduced the unit price of one of its products by , the number of units sold increased by . As a result, the total revenue from the product rose by $2,400 compared with the revenue before the price change. What was the manufacturer's total revenue, in dollars, before the price change?
A changed unit price and a changed number sold affect revenue together. Use percent multipliers for each change, then multiply them because revenue is price times units sold. Work backward from the dollar increase with a one-unknown regression in Desmos. Don't subtract the two percentages: they describe changes to different factors.
Hints
- Hint 1
A percent decrease tells you what was removed, not what remains. Start with the whole original price, or , and take away the reduction. What price multiplier is left?
- Hint 2
Revenue is unit price times units sold. The increase in units gives its own multiplier. What happens to a product when you multiply one factor by the price multiplier and the other by the units multiplier?
- Hint 3
The $2,400 is the increase in revenue, not the new total. If is the old revenue, write the new revenue using your combined multiplier. Then subtract .
Step-by-step
Multiply the changes, then solve for the original
Step 1Find the price multiplier
A decrease leaves the part that wasn't removed. The price keeps of its old value, so its multiplier, the number applied to the old price, is:
- Step 2
Find the units-sold multiplier
An increase keeps the whole original count and adds . So the new number of units sold has multiplier :
- Step 3
Combine the multipliers for revenue
Revenue is unit price times units sold. Because both factors change, multiply their multipliers to get the revenue multiplier. Type in Desmos. It shows , so the new revenue is times the old revenue.
- Step 4
Write an equation for the increase
Let be the old revenue in dollars. The new revenue is . The increase is new minus old, so the $2,400 rise gives:
Don't set equal to : that would treat the increase as the new total.
- Step 5
Solve for the old revenue
Type in Desmos. Use instead of and instead of so Desmos solves for the unknown rather than graphing the equation. Under PARAMETERS, it reports . The manufacturer's revenue before the price change was $20,000. Grid in 20000.