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Algebra

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At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to \$3.15. Gary orders a bagel and a muffin, which comes out to \$3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to \$8.15. How many cents does one bagel cost?

Jun 1, 2024

#1
+850
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At a cafeteria,

Mary orders two pieces of toast and a bagel, which comes out to \$3.15.

Gary orders a bagel and a muffin, which comes out to \$3.50.

Larry orders a piece of toast, two bagels, and three muffins, which comes out to \$8.15.

How many cents does one bagel cost?

Mary          2T + B = 315 ............. —>  T = (315 – B) / 2

Gary          B + M = 350 ................—>  M = (350 – B)

Larry          T + 2B + 3M = 815

Substitute for Toast & Muffin into Larry's order

[(315 – B) / 2]  +  2B  +  [3 (350 – B)]  =  815

[(315 – B) / 2]  +  2B  +  (1050 – 3B)  =  815

Multiply both sides by 2       (315 – B)  +  4B  +  (2100 – 6B)  =  1630

Combine like terms              –3B + 2415  =  1630

–3B  =  – 785

B  =  261 2/3  ¢

They usually construct these problems to come out with an even number.

I've gone back over the steps a hundred times and cannot find a mistake.

.

Jun 2, 2024
#2
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From the problem, we can write the system

\(2T + 1B = 3.15 \\ 1M + 1B = 3.50 \\ 1T + 2B + 3M = 8.15 \)

where t is toast, b is bagel and m is muffin.

From the first two equations, we have

\( [3.15 - B ] / 2 = T \\ M = 3.50 - B\)

Subbing this in to the third equation, we have

\([ 3.15 - B ] / 2 + 2B + 3 [ 3.50 - B ] = 8.15 \\ [3.15 - B ] + 4B + 6 [ 3.50 -B ] = 16.30 \\ -3B + 3.15 + 21 = 16.30 \\ -3B = 16.30 - 3.15 - 21 \\ -3B = -7.85 \\ B = 7.85 / 3 ≈ \$2.62\)