Gassing up at the station

It s no easy choice when you pull up at the gas station to fill up your vehicle. First, you have to take a deep breath about the price, and then you have to choose between regular gas, premium gas, or even gas that contains ethanol. 

One of the equations you need deals with cost, and the other deals with the number of miles. The common element in both equations is the number of gallons of each type of gas — and the number of gallons answers the question, too. Let r represent the number of gallons of regular gas and p represent the number of gallons of premium gas. The total cost, 104.40 = 2.70r + 3.15p. The total number of miles, 748 = 19r + 23p. None of the coefficients of the variables is equal to 1, so you have to make a choice as to which variable to solve for. Because the coefficient 19 is the smallest number, 1 opt to solve for r in the second equation and replace the r in the first equation with that equivalence in terms of p. 

Distributing the 2.70 and simplifying don t make for very pretty computations, but a calculator makes short work of all the operations. I choose to find a common denominator to combine the fractions and decimals, because the decimal you get with a denominator of 19 just keeps repeating. Then you solve for p by multiplying each side of the equation by the reciprocal of its coefficient. 

Stefanie used 16 gallons of premium gas. Substitute the 16 for p in the equation 19r + 23p = 748, and you get 19r + 23(16) = 748. This simplifies to 19r + 368 = 748. Subtracting 368 from each side, the equation becomes 19r = 380. Dividing each side by 19, you get that r = 20. She bought 20 gallons of regular gas. 

The Problem A service group is selling candy bars and bags of almonds for a fund-raiser. They re selling the candy bars (which cost them 40 cents each) for 1 and the bags of almonds (which cost them 50 t each) for 1.25. Their total receipts (revenue) for the sale of 1,350 items was 1,500. How many of each item did they sell, and what was their profit  

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