Sadie and her children went into a grocery store where they sell peaches for $\$ 1.25$ each and mangos for $\$ 0.50$ each. Sadie has $\$ 9.50$ to spend and must buy no less than $13$ peaches and mangos altogether. If Sadie decided to buy $4$ mangos, determine all possible values for the number of peaches that she could buy. $\newline$ Answer:

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Q. Sadie and her children went into a grocery store where they sell peaches for

\$ 1.25

each and mangos for

\$ 0.50

each. Sadie has

\$ 9.50

to spend and must buy no less than

13

peaches and mangos altogether. If Sadie decided to buy

4

mangos, determine all possible values for the number of peaches that she could buy.

\newline

Answer:

Calculate Cost of Mangos: Calculate the total cost of $4$ mangos. $\newline$ Since each mango costs $\$0.50$ , we multiply the number of mangos by the cost per mango. $\newline$ $4$ mangos $*$ $\$0.50$ /mango = $\$2.00$
Subtract Mango Cost: Subtract the cost of the mangos from the total amount Sadie has to spend. $\newline$ Sadie has $\$9.50$ to spend, and she has already spent $\$2.00$ on mangos. $\newline$ $\$9.50 - \$2.00 = \$7.50$ $\newline$ This is the amount remaining for purchasing peaches.
Calculate Maximum Peaches: Calculate the maximum number of peaches Sadie can buy with the remaining money. $\newline$ Since each peach costs $\$1.25$ , we divide the remaining amount by the cost per peach. $\newline$ $\$7.50 / \$1.25/\text{peach} = 6$ peaches
Check Total Fruits: Check if the total number of fruits meets the minimum requirement. $\newline$ Sadie has bought $4$ mangos and can buy up to $6$ peaches. $\newline$ $4$ mangos + $6$ peaches = $10$ fruits $\newline$ This does not meet the requirement of buying at least $13$ fruits.
Determine Peaches Needed: Determine the number of peaches Sadie can buy to meet the minimum fruit requirement. $\newline$ Sadie needs to buy at least $13 - 4 = 9$ peaches to meet the minimum requirement of $13$ fruits.
Check Affordability: Check if Sadie has enough money to buy $9$ peaches. $\newline$ $9$ peaches $\times$ $\$$ $1$ . $25$ /peach = $\$$ $11$ . $25$ $\newline$ Sadie only has $\$$ $7$ . $50$ left to spend on peaches, so she cannot buy $9$ peaches.
Find Maximum Peaches: Find the greatest number of peaches Sadie can buy with the remaining money that also meets the minimum fruit requirement. $\newline$ Sadie can buy $6$ peaches with her remaining $\$7.50$ , but this does not meet the minimum requirement of $13$ fruits. Therefore, she must buy fewer mangos or find a different solution.
Re-evaluate Constraints: Since the previous steps led to a contradiction, we need to re-evaluate our calculations and constraints. $\newline$ We need to find a number of peaches that Sadie can afford with $\$7.50$ that, when added to the $4$ mangos, equals or exceeds $13$ fruits in total. $\newline$ Let's denote the number of peaches as $P$ . $\newline$ $P + 4$ mangos $\geq 13$ fruits $\newline$ $P \geq 13 - 4$ $\newline$ $P \geq 9$
Adjust Mango Quantity: Calculate the cost for $9$ peaches. $\newline$ $9$ peaches $\times \$1.25/\text{peach} = \$11.25$ $\newline$ This exceeds Sadie's remaining budget of $\$7.50$ , so she cannot buy $9$ peaches.
Calculate Peaches Affordability: Adjust the number of peaches to fit the budget. $\newline$ Since Sadie cannot afford $9$ peaches, let's try with $8$ peaches. $\newline$ $8$ peaches * $\$1.25/\text{peach}$ = $\$10.00$ $\newline$ This still exceeds Sadie's remaining budget of $\$7.50$ .
Check Fruit Requirement: Continue adjusting the number of peaches. $\newline$ Try with $7$ peaches. $\newline$ $7$ peaches $\times$ $\$$ $1$ . $25$ /peach = $\$$ $8$ . $75$ $\newline$ This still exceeds Sadie's remaining budget of $\$$ $7$ . $50$ .
Adjust Mango Quantity: Continue adjusting the number of peaches. $\newline$ Try with $6$ peaches. $\newline$ $6$ peaches $*$ $\$1.25/$ peach = $\$7.50$ $\newline$ This fits Sadie's remaining budget exactly, and when added to the $4$ mangos, gives a total of $10$ fruits.
Calculate Peaches Affordability: Since Sadie needs at least $13$ fruits, and $6$ peaches and $4$ mangos only make $10$ fruits, she cannot buy $6$ peaches. She needs to buy at least $9$ fruits to meet the minimum requirement when combined with the $4$ mangos. $\newline$ Since Sadie cannot afford $9$ peaches, she cannot meet both the budget constraint and the minimum fruit requirement if she buys $4$ mangos.
Check Fruit Requirement: Re-evaluate the constraints and find a solution that meets both the budget and the minimum fruit requirement. $\newline$ Since Sadie cannot meet the minimum fruit requirement by buying $4$ mangos and the maximum number of peaches she can afford, she needs to buy fewer mangos to be able to buy more peaches.
Adjust Mango Quantity: Determine the new number of mangos Sadie can buy to afford more peaches. $\newline$ If Sadie buys $3$ mangos instead of $4$ , she will spend: $\newline$ $3$ mangos $\times$ $\$$ $0$ . $50$ /mango = $\$$ $1$ . $50$ $\newline$ This leaves her with: $\newline$ $\$$ $9$ . $50$ - $\$$ $1$ . $50$ = $\$$ $8$ . $00$ for peaches.
Calculate Peaches Affordability: Calculate the number of peaches Sadie can buy with $\$8.00$ . $\$8.00 / \$1.25/\text{peach} = 6.4 \text{peaches}$ Since Sadie cannot buy a fraction of a peach, she can buy up to $6$ peaches with $\$8.00$ .
Check Fruit Requirement: Check if the new combination of $3$ mangos and $6$ peaches meets the minimum fruit requirement. $\newline$ $3$ mangos + $6$ peaches = $9$ fruits $\newline$ This still does not meet the minimum requirement of $13$ fruits.
Adjust Mango Quantity: Since buying $3$ mangos still does not allow Sadie to meet the minimum fruit requirement, she needs to buy even fewer mangos. $\newline$ If Sadie buys $2$ mangos, she will spend: $\newline$ $2$ mangos $\times$ $\$$ $0$ . $50$ /mango = $\$$ $1$ . $00$ $\newline$ This leaves her with: $\newline$ $\$$ $9$ . $50$ - $\$$ $1$ . $00$ = $\$$ $8$ . $50$ for peaches.
Calculate Peaches Affordability: Calculate the number of peaches Sadie can buy with $\$8.50$ . $\$8.50 / \$1.25/\text{peach} = 6.8$ peaches Since Sadie cannot buy a fraction of a peach, she can buy up to $6$ peaches with $\$8.50$ .
Check Fruit Requirement: Check if the new combination of $2$ mangos and $6$ peaches meets the minimum fruit requirement. $\newline$ $2$ mangos + $6$ peaches = $8$ fruits $\newline$ This still does not meet the minimum requirement of $13$ fruits.
Adjust Mango Quantity: Since buying $2$ mangos still does not allow Sadie to meet the minimum fruit requirement, she needs to buy even fewer mangos. $\newline$ If Sadie buys $1$ mango, she will spend: $\newline$ $1$ mango $\times$ $\$$ $0$ . $50$ /mango = $\$$ $0$ . $50$ $\newline$ This leaves her with: $\newline$ $\$$ $9$ . $50$ - $\$$ $0$ . $50$ = $\$$ $9$ . $00$ for peaches.
Calculate Peaches Affordability: Calculate the number of peaches Sadie can buy with $\$9.00$ . $\$9.00 / \$1.25/\text{peach} = 7.2 \text{peaches}$ Since Sadie cannot buy a fraction of a peach, she can buy up to $7$ peaches with $\$9.00$ .
Check Fruit Requirement: Check if the new combination of $1$ mango and $7$ peaches meets the minimum fruit requirement. $\newline$ $1$ mango + $7$ peaches = $8$ fruits $\newline$ This still does not meet the minimum requirement of $13$ fruits.
No Solution: Since buying $1$ mango still does not allow Sadie to meet the minimum fruit requirement, she needs to buy no mangos and all peaches. $\newline$ With $\$9.50$ to spend on peaches alone: $\newline$ $\$9.50 / \$1.25/\text{peach} = 7.6$ peaches $\newline$ Sadie can buy up to $7$ peaches with $\$9.50$ .
No Solution: Since buying $1$ mango still does not allow Sadie to meet the minimum fruit requirement, she needs to buy no mangos and all peaches. $\newline$ With $\$9.50$ to spend on peaches alone: $\newline$ $\$9.50 / \$1.25/\text{peach} = 7.6$ peaches $\newline$ Sadie can buy up to $7$ peaches with $\$9.50$ .Check if buying $7$ peaches meets the minimum fruit requirement. $\newline$ $7$ peaches = $7$ fruits $\newline$ This does not meet the minimum requirement of $13$ fruits.
No Solution: Since buying $1$ mango still does not allow Sadie to meet the minimum fruit requirement, she needs to buy no mangos and all peaches. $\newline$ With $\$9.50$ to spend on peaches alone: $\newline$ $\$9.50 / \$1.25/\text{peach} = 7.6$ peaches $\newline$ Sadie can buy up to $7$ peaches with $\$9.50$ .Check if buying $7$ peaches meets the minimum fruit requirement. $\newline$ $7$ peaches = $7$ fruits $\newline$ This does not meet the minimum requirement of $13$ fruits.Since Sadie cannot meet the minimum requirement of $13$ fruits with her budget, there is no solution to the problem as stated. Sadie cannot buy $\$9.50$ $0$ mangos and at least $13$ fruits in total with $\$9.50$ .