Farah wants to buy a television and a radio for her grandparents. On an Internet sales, the price of a television to a radio is $7 : 4$ . After a discount of $\$132$ on each item, the price ratio of the television to radio becomes $25 : 13$ . What is the discounted price of the television?

Full solution

Q. Farah wants to buy a television and a radio for her grandparents. On an Internet sales, the price of a television to a radio is

7 : 4

. After a discount of

\$132

on each item, the price ratio of the television to radio becomes

25 : 13

. What is the discounted price of the television?

Denote Prices: Let's denote the original price of the television as \$$7$x\ and the original price of the radio as \$$4$x\, where $x$ is a common multiplier. The ratio of their prices is $7:4$.
Apply Discount: After a discount of $\$132$ on each item, the price of the television becomes $\$7x - \$132$ and the price of the radio becomes $\$4x - \$132$.
Calculate New Ratio: The new ratio of the prices after the discount is given as $25:13$. This means that the discounted price of the television to the discounted price of the radio is $25:13$.
Set Up Proportion: We can set up a proportion to find the new prices: $(7x - 132)/ (4x - 132) = \frac{25}{13}$.
Cross-Multiply: Cross-multiply to solve for $x$: $13(7x - 132) = 25(4x - 132)$.
Simplify Equation: This simplifies to $91x - 1716 = 100x - 3300$.
Subtract Terms: Subtract $91x$ from both sides to get: $-1716 = 9x - 3300$.
Add Terms: Add $3300$ to both sides to solve for $x$: $1584 = 9x$.
Calculate $x$: Divide both sides by $9$ to find $x$: $x = \frac{1584}{9}$.
Find Original Price: Calculating $x$ gives us: $x = 176$.
Find Discounted Price: Now we can find the original price of the television, which is $7x$: $7 \times 176 = \$(1232)$.
Find Discounted Price: Now we can find the original price of the television, which is $7x$: $7 \times 176 = \$(1232)$.Finally, we subtract the discount from the original price of the television to find the discounted price: $\$(1232) - \$(132) = \$(1100)$.