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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 . After a discount of on each item, the price ratio of the television to radio becomes . What is the discounted price of the television?
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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 . After a discount of on each item, the price ratio of the television to radio becomes . 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: \(\$(1232) - \$(132) = \$(1100)\).
