Algebra 1 word problems help please
1. Madison is making a vinaigrette for her catering service using olive oil that costs 50 cents per ounce and apple cider vinegar that costs 35 cents an ounce.The ideal batch of vinaigrette has 25% vinegar.If she has already combined 3 cups of vinegar and 1 cup of oil,how many cups of oil should she add? What is the total cost of her completed recipe?
2. The projected cost of a new high school is $20,066,340, of which $5,216,340 will be covered by available grants. Using data from the last census, the planning committee estimated the cost per town resident. However, new census data reveals that the population has increased by 2,200 people, reducing the cost per town resident by $75. What is the current population of the town?
a 25% vinegar solution is 1/4 vinegar. So, if she has 3c vinegar, she will need 12c of solution.
cost to city is 20066340-5216340 = 14,850,000
So, if the old population was p,
14850000/(p+2200) = 14850000/p - 75
1. To solve the first problem, we need to find out how many cups of oil Madison should add and the total cost of the recipe.
Step 1: Find the total volume of the vinaigrette mixture:
Since 3 cups of vinegar and 1 cup of oil have already been combined, the total volume of the vinaigrette is 3 + 1 = 4 cups.
Step 2: Determine the ideal ratio of vinegar and oil:
The ideal batch has 25% vinegar, which means that the vinegar should make up 25% of the total volume. Using proportions, we can set up the equation: 3 cups / 4 cups = 25% / 100%.
Step 3: Calculate the desired amount of oil:
To find the amount of oil needed, we subtract the amount of vinegar from the total volume: 4 cups - 3 cups = 1 cup. Madison should add 1 cup of oil.
Step 4: Calculate the total cost:
To calculate the total cost, we need to know the cost per cup of both vinegar and oil.
- Vinegar cost: 3 cups * 35 cents/ounce = (3 * 8) ounces * 35 cents/ounce = 24 ounces * 35 cents/ounce = 840 cents.
- Oil cost: 1 cup * 50 cents/ounce = 8 ounces * 50 cents/ounce = 400 cents.
The total cost is the sum of the vinegar and oil costs: 840 cents + 400 cents = 1240 cents, or $12.40.
Therefore, Madison should add 1 cup of oil, and the total cost of her completed recipe is $12.40.
2. To solve the second problem, we need to find the current population of the town.
Step 1: Calculate the initial cost per resident:
The projected cost of the new high school is given as $20,066,340, and it will be covered by $5,216,340 in grants. So, the remaining cost is $20,066,340 - $5,216,340 = $14,850,000.
Step 2: Determine the number of residents at the time of the last census:
We don't have this information directly, but we are given that the planning committee used data from the last census to estimate the cost per resident. This means that the cost per resident at that time was $14,850,000 divided by the population at that time.
Step 3: Find the reduction in cost per resident due to the population increase:
We are told that the current population has increased by 2,200 people, reducing the cost per resident by $75. Therefore, we can set up the equation: $75 * 2,200 = $14,850,000 / (current population - 2,200).
Step 4: Solve for the current population:
Using this equation, we can rearrange it to solve for the current population: $14,850,000 = $75 * 2,200 + ($75 * current population) - ($75 * 2,200).
Simplifying further, we have: $14,850,000 = $165,000 + $75 * current population - $165,000.
Combining like terms, we get: $14,850,000 = $75 * current population.
Dividing both sides of the equation by $75, we find: current population = $14,850,000 / $75.
Using a calculator, we can calculate the current population by dividing $14,850,000 by $75. The result will be the current population of the town.