CEMC Hypatia, 2020

The cost of 12 bags of avocados is $\$5.00\times12=\$60.00$.

Thus, the chef spent $\$135.00-\$60.00=\$75.00$ on mangoes.
The cost of each box of mangoes is $\$12.50$, and so the chef purchased $\dfrac{\$75.00}{\$12.50}=6$ boxes of mangoes.
Solution 1

A bag of avocados sells for $5.00, and so a 10% discount represents a savings of $\frac{10}{100}\times\$5.00$ or $0.10\times\$5.00$ which equals $\$0.50$.

Thus, the discounted price for a bag of avocados is $\$5.00-\$0.50=\$4.50$.

A box of mangoes sells for $12.50, and so a 20% discount represents a savings of $\frac{20}{100}\times\$12.50$ or $0.20\times\$12.50$ which equals $\$2.50$.

Thus, the discounted price for a box of mangoes is $\$12.50-\$2.50=\$10.00$.
On Saturdays, the total cost for 8 bags of avocados and 4 boxes of mangoes is $$\$4.50\times8+\$10.00\times4=\$36.00+\$40.00=\$76.00$$

Solution 2

A bag of avocados sells for $5.00, and so a 10% discount is equivalent to paying $100\%-10\%=90\%$ of the regular price.

Thus, the discounted price for a bag of avocados is $\frac{90}{100}\times\$5.00=0.90\times\$5.00=\$4.50$.

A box of mangoes sells for $12.50, and so a 20% discount is equivalent to paying $100\%-20\%=80\%$ of the regular price.

Thus, the discounted price for a box mangoes is $\frac{80}{100}\times\$12.50=0.80\times\$12.50=\$10.00$.
On Saturdays, the total cost for 8 bags of avocados and 4 boxes of mangoes is $$\$4.50\times8+\$10.00\times4=\$36.00+\$40.00=\$76.00$$
Avocados are sold in bags of 6 and the chef needs 100 avocados.

Since $6\times16=96$ and $6\times17=102$, the chef will need to purchase 17 bags of avocados (16 bags is not enough).

Mangoes are sold in boxes of 15 and the chef needs 70 mangoes.

Since $15\times4=60$ and $15\times5=75$, the chef will need to purchase 5 boxes of mangoes.
The total cost of the purchase was $\$5.00\times17+\$12.50\times5=\$85.00+\$62.50=\$147.50$.
Avocados are sold for $5.00 per bag, and so the cost to purchase any number of bags is a whole number.

Mangoes are sold for $12.50 per box, and so the cost to purchase mangoes is a whole number only when an even number of boxes are bought (1 box costs $12.50, 2 boxes cost $25.00, 3 boxes cost $37.50, and so on).

Since the chef spends exactly $75.00 (a whole number), then the chef must purchase an even number of boxes of mangoes.

If the chef purchases 2 boxes of mangoes, the cost is $25.00, which leaves $\$75.00-\$25.00=\$50.00$ to be used to purchase $\$50.00\div \$5.00=10$ bags of avocados.

In this case, the chef has $10\times6=60$ avocados and $2\times15=30$ mangoes.

Each tart requires 1 avocado and 2 mangoes and so the chef can make $30\div2=15$ tarts (he has more than 15 avocados but only 30 mangoes).

If the chef purchases 4 boxes of mangoes, the cost is $4\times\$12.50=\$50.00$, which leaves $\$75.00-\$50.00=\$25.00$ to be used to purchase $\$25.00\div \$5.00=5$ bags of avocados.

In this case, the chef has $5\times6=30$ avocados and $4\times15=60$ mangoes.

Each tart requires 1 avocado and 2 mangoes and so the chef can make 30 tarts.

If the chef purchases 6 boxes of mangoes, the cost is $6\times\$12.50=\$75.00$, which leaves no money to purchase avocados.

Purchasing more than 6 boxes of mangoes will cost the chef more than $75.00.
Therefore, if the chef purchases 30 avocados (5 bags) and 60 mangoes (4 boxes), she will have spent exactly $75.00, have twice as many mangoes as avocados, and be able to make the greatest number of tarts, 30.