Question

Isaac paid $2.75 for 4 granola bars and 1 apple. Amelia paid $2.25 for 2 granola bars and 3 apples. Find the cost of a granola bar and an apple.

Answer 1

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: QUESTION \: \: } \mid}}}}}$

$\: \: \: \: \text{Isaac paid \ 2.75 for 4 granola bars and} \\ \text{ 1 apple. Amelia paid \ 2.25 for 2 granola } \\ \text{bars and 3 apples. Find the cost of a } \\ \text{granola bar and an apple.}$

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{\mathfrak{ \: \: Given, \: \: }} \\ \\ \pink{ \text{Amount paid for 4 granola bars and} } \\ \: \: \: \: \pink{\text{1 apple = \ 2.75}} \\ \\ \pink{ \text{Amount paid for 2 granola bars and }} \\ \: \: \: \: \pink{\text{3 apple = \ 2.22}} \\ \\ \\ \underline{ \mathfrak{ \: \: To \: \: find : \mapsto \: }} \\ \\ \: \: \: \: \green{\text{Cost of a granola bar = ?} } \\ \: \: \: \: \green{\text{Cost of an apple = ?}} \\ \\ \\ \underline{ \mathfrak{ \: \: Suppose, \: \: }} \\ \\ \: \: \: \: \: \: \red{\text{Cost of a granola bar = \ x }} \\ \tt{And,} \\ \: \: \: \: \: \red{ \text{ Cost of an apple = \ y}}$

$\underline{\text{ According to first condition \: }} \\ \\ \pink{\text{Amount paid for 4 granola bars and }} \\ \: \: \: \: \: \pink{\text{1 apple = \ 2.75}} \\ \\ \tt{ : \leadsto : \: \: 4x + y = 2.75 \: \: - - - - - > (1)} \\ \\ \underline{ \text{ \: According to second condition \: }} \\ \\ \pink {\text{Amount paid for 2 granola bars and}} \\ \: \: \: \: \pink{ \text{ 3 apple = \ 2.22} }\\ \\ \tt{ : \leadsto : \: 2x + 3y = 2.22 \: \: - - - -- > (2)}$

$\mathfrak{Now,} \\ \\ \tt {(2) \times 2 \implies \: 2 \times 2x + 2 \times 3y = 2 \times 2.22 }\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt {\implies \: 4x + 6y = 4.44 \: \: - - - - - > (3)} \\ \\ \tt{ \therefore \:(3) - (1) \implies \: 5y = 1.69 } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \implies \: y = \frac{1.69}{5} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{ \therefore \: \: y = 0.338 \: }$

$\underline{ \text{ \: \: Putting the value of \: \red{y} \: in eq. (1) , we get \: \: }} \\ \\ \tt{(1) \implies \:4x + y = 2.75 } \\ \\ \: \: \: \: \: \: \: \tt{ \implies \: 4x + 0.338 = 2.75} \\ \\\: \: \: \: \: \: \: \tt{ \implies \: 4x = 2.75 - 0.338} \\ \\\: \: \: \: \: \: \: \tt{ \implies \:4x = 2.412 }\\ \\ \: \: \: \: \: \: \:\tt{ \implies \:x = \frac{2.412}{4}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \:\tt{ \therefore \: \: x = 0.603}$

$\therefore \: \red{\text{Cost of a granola bar = \ 0.603 }} \\ \tt{And,} \\ \: \: \: \: \: \red{ \text{ Cost of an apple = \ 0.338}}$

$\huge{ \underline{ \overline{ \mid{ \mathfrak{ \purple{ \: \: VERIFICATION \: \: } \mid}}}}}$

$\underline{ \mathfrak{ \: \: We \: \: have, \: \: }} \\ \\ \red{ \text{Cost of a granola bar = \ 0.603}} \\ \\ \red{\text{Cost of an apple = \ 0.338}}$

In case of Issac,

$\tt{Cost \: \: of \: \: 4 \: \: granola \: \: bars = \ (0.603 \times 4)} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{= \ 2.412} \\ \\ \tt{Cost \: \: of \: \: 1 \: \: apple = \ (0.338 \times 1) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{= \ 0.338} \\ \\ \therefore \: \: \text{Total amount paid = \ (2.412 + 0.338) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{= \ 2.75}$

In case of Amelia,

$\tt{Cost \: \: of \: \: 2 \: \: granola \: \: bars = \ (0.603 \times 2) } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{= \ 1.206} \\ \\ \tt{Cost \: \: of \: \: 3 \: \: apples = \ (0.338 \times 3) } \\ \\\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt{= \ 1.014} \\ \\ \therefore \: \: \text{Total amount paid = \ (2.412 + 0.338)} \\ \\\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \text{ = \ 2.22}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bold{ \underline{ \: \: Hence, Verified. \: \: }}$

Answer 2

$\huge{ \underline{ \overline{ \mid{ \mathbb{ \red{ \: \: SOLUTION \: \: } \mid}}}}}$

$\underline{\bold{ \: \: Given, \: \: }} \\ \\ \text{Amount paid for 4 granola bars and} \\ \: \: \: \: \text{1 apple = \ 2.75}\\ \\ \text{Amount paid for 2 granola bars and } \\ \: \: \: \: \text{3 apple = \ 2.22} \\ \\ \\ \underline{ \bold{ \: \: To \: \: find : \to \: }} \\ \\ \: \: \: \: \text{Cost of a granola bar = ?} \\ \: \: \: \: \text{Cost of an apple = ?}\\ \\ \\ \underline{ \bold{ \: \: Suppose, \: \: }} \\ \\ \: \: \: \: \: \: \text{Cost of a granola bar = \ x } \\ \bf{And,} \\ \: \: \: \: \: \text{ Cost of an apple = \ y}$

$\underline{\text{ According to first condition \: }} \\ \\ \text{Amount paid for 4 granola bars and } \\ \: \: \: \: \: \text{1 apple = \ 2.75} \\ \\ \bf{ : \to : \: \: 4x + y = 2.75 \: \: - - - - - > (1)} \\ \\ \underline{ \text{ \: According to second condition \: }} \\ \\ \text{Amount paid for 2 granola bars and} \\ \: \: \: \: \text{ 3 apple = \ 2.22} \\ \\ \bf{ : \to : \: 2x + 3y = 2.22 \: \: - - - -- > (2)}$

$\mathfrak{Now,} \\ \\ \bf{(2) \times 2 \implies \: 2 \times 2x + 2 \times 3y = 2 \times 2.22 }\\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf {\implies \: 4x + 6y = 4.44 \: \: - - - - - > (3)} \\ \\ \bf{ \therefore \:(3) - (1) \implies \: 5y = 1.69 } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{ \implies \: y = \frac{1.69}{5} } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \bf{ \therefore \: \red{ y = 0.338 \: }}$

$\underline{ \text{ \: \: Substituting the value of \: \red{y} \: in eq. (1) , we get \: \: }} \\ \\ \bf{(1) \implies \:4x + y = 2.75 } \\ \\ \: \: \: \: \: \: \: \bf{ \implies \: 4x + 0.338 = 2.75} \\ \\\: \: \: \: \: \: \: \bf{ \implies \: 4x = 2.75 - 0.338} \\ \\\: \: \: \: \: \: \: \bf{ \implies \:4x = 2.412 }\\ \\ \: \: \: \: \: \: \:\bf{ \implies \:x = \frac{2.412}{4}} \\ \\ \: \: \: \: \: \: \: \: \: \: \: \:\bf{ \therefore \: \red{ x = 0.603}}$

$\therefore \: \text{Cost of a granola bar = \ 0.603 } \\ \sf{And,} \\ \: \: \: \: \: \text{ Cost of an apple = \ 0.338}$