phase 2: medium levels (questions 7-14)
focus: unit price comparisons, three-term ratios, and multi-step rate calculations.

7. brand a grass seed costs \$24 for a $4\text{ kg}$ bag. brand b grass seed costs \$35 for a $5\text{ kg}$ bag. calculate the unit rate for both to determine which brand is the better buy.

8. a specific concrete mix uses a three-term ratio of cement to sand to gravel of $1 : 2 : 3$ by volume. if a construction project requires $12\text{ m}^3$ of gravel, what is the total volume of concrete mix needed?

9. a driver uses $18\text{ liters}$ of fuel to travel $200\text{ km}$. at this exact rate, how many liters of fuel will they need to drive a distance of $500\text{ km}$?

10. on a detailed regional map, the scale ratio is given as $1 : 50,000$. if a local bike trail measures $8\text{ cm}$ on the map, what is its actual real-world length in kilometers (km)?

11. a student can type $120\text{ words}$ in $3\text{ minutes}$. at this identical unit rate, how many words can they type in $7\text{ minutes}$?

12. a store sells laundry detergent in two sizes: a $2\text{ l}$ bottle for \$7.50 and a $5\text{ l}$ bottle for \$18.00. calculate the unit price per liter for both sizes and state which one gives you more value for your money.

13. the ratio of the width to the length of a rectangular backyard patio is $4 : 7$. if the actual length of the patio is $14\text{ meters}$, what is the perimeter of the patio?

14. a standard tap drips water at a constant rate of $15\text{ ml}$ per minute. how many liters of water are wasted if the tap drips continuously for exactly $2\text{ hours}$? (remember: $1\text{ l} = 1000\text{ ml}$).

Question

phase 2: medium levels (questions 7-14)
focus: unit price comparisons, three-term ratios, and multi-step rate calculations.

7. brand a grass seed costs \$24 for a $4\text{ kg}$ bag. brand b grass seed costs \$35 for a $5\text{ kg}$ bag. calculate the unit rate for both to determine which brand is the better buy.

8. a specific concrete mix uses a three-term ratio of cement to sand to gravel of $1 : 2 : 3$ by volume. if a construction project requires $12\text{ m}^3$ of gravel, what is the total volume of concrete mix needed?

9. a driver uses $18\text{ liters}$ of fuel to travel $200\text{ km}$. at this exact rate, how many liters of fuel will they need to drive a distance of $500\text{ km}$?

10. on a detailed regional map, the scale ratio is given as $1 : 50,000$. if a local bike trail measures $8\text{ cm}$ on the map, what is its actual real-world length in kilometers (km)?

11. a student can type $120\text{ words}$ in $3\text{ minutes}$. at this identical unit rate, how many words can they type in $7\text{ minutes}$?

12. a store sells laundry detergent in two sizes: a $2\text{ l}$ bottle for \$7.50 and a $5\text{ l}$ bottle for \$18.00. calculate the unit price per liter for both sizes and state which one gives you more value for your money.

13. the ratio of the width to the length of a rectangular backyard patio is $4 : 7$. if the actual length of the patio is $14\text{ meters}$, what is the perimeter of the patio?

14. a standard tap drips water at a constant rate of $15\text{ ml}$ per minute. how many liters of water are wasted if the tap drips continuously for exactly $2\text{ hours}$? (remember: $1\text{ l} = 1000\text{ ml}$).

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Ratios", "Unit Rates", "Equivalent Ratios" ], "new_concepts": [ "Three-Term Ratios", "Scale Conversions", "Rate Conversions" ], "current_concepts": [ "Unit Rates", "Ratios", "Equivalent Ratios", "Three-Term Ratios", "Scale Conversions", "Rate Conversions" ] } ## Question 7: Calculate unit rates and compare $$ \begin{aligned} &\text{Brand A: } \frac{\$24}{4\text{ kg}} = \$6.00\text{ per kg}\ &\text{Brand B: } \frac{\$35}{5\text{ kg}} = \$7.00\text{ per kg}\ &\$6.00 < \$7.00 \implies \text{Brand A is the better buy.} \end{aligned} $$ ## Question 8: Determine total volume of concrete mix $$ \begin{aligned} &\text{Ratio (cement : sand : gravel)} = 1 : 2 : 3\ &\text{Total parts} = 1 + 2 + 3 = 6\ &\text{Gravel volume} = 12\text{ m}^3 \implies 3\text{ parts} = 12\text{ m}^3 \implies 1\text{ part} = 4\text{ m}^3\ &\text{Total volume} = 6 \times 4\text{ m}^3 = 24\text{ m}^3 \end{aligned} $$ ## Question 9: Calculate fuel needed for 500 km $$ \begin{aligned} &\text{Fuel rate} = \frac{18\text{ L}}{200\text{ km}} = 0.09\text{ L/km}\ &\text{Fuel for 500 km} = 0.09\text{ L/km} \times 500\text{ km} = 45\text{ L} \end{aligned} $$ ## Question 10: Convert map scale to actual distance $$ \begin{aligned} &\text{Actual distance in cm} = 8\text{ cm} \times 50,000 = 400,000\text{ cm}\ &\text{Actual distance in meters} = \frac{400,000\text{ cm}}{100\text{ cm/m}} = 4,000\text{ m}\ &\text{Actual distance in km} = \frac{4,000\text{ m}}{1,000\text{ m/km}} = 4\text{ km} \end{aligned} $$ ## Question 11: Calculate words typed in 7 minutes $$ \begin{aligned} &\text{Typing rate} = \frac{120\text{ words}}{3\text{ minutes}} = 40\text{ words/minute}\ &\text{Words in 7 minutes} = 40\text{ words/minute} \times 7\text{ minutes} = 280\text{ words} \end{aligned} $$ ## Question 12: Calculate unit prices and compare value $$ \begin{aligned} &\text{2 L bottle: } \frac{\$7.50}{2\text{ L}} = \$3.75\text{ per L}\ &\text{5 L bottle: } \frac{\$18.00}{5\text{ L}} = \$3.60\text{ per L}\ &\$3.60 < \$3.75 \implies \text{The 5 L bottle gives more value.} \end{aligned} $$ ## Question 13: Calculate perimeter of the patio $$ \begin{aligned} &\text{Ratio (width : length)} = 4 : 7\ &\text{Length} = 14\text{ m} \implies 7\text{ parts} = 14\text{ m} \implies 1\text{ part} = 2\text{ m}\ &\text{Width} = 4 \times 2\text{ m} = 8\text{ m}\ &\text{Perimeter} = 2 \times (\text{width} + \text{length}) = 2 \times (8\text{ m} + 14\text{ m}) = 44\text{ m} \end{aligned} $$ ## Question 14: Calculate wasted water in liters $$ \begin{aligned} &\text{Total minutes} = 2\text{ hours} \times 60\text{ minutes/hour} = 120\text{ minutes}\ &\text{Total mL wasted} = 15\text{ mL/minute} \times 120\text{ minutes} = 1,800\text{ mL}\ &\text{Total Liters wasted} = \frac{1,800\text{ mL}}{1,000\text{ mL/L}} = 1.8\text{ L} \end{aligned} $$ ## Question 7 Brand A unit rate: $\$6.00\text{ per kg}$ Brand B unit rate: $\$7.00\text{ per kg}$ Better buy: Brand A ## Question 8 Total volume of concrete mix: $24\text{ m}^3$ ## Question 9 Fuel needed: $45\text{ Liters}$ ## Question 10 Actual length: $4\text{ km}$ ## Question 11 Words typed: $280\text{ words}$ ## Question 12 2 L bottle unit price: $\$3.75\text{ per Liter}$ 5 L bottle unit price: $\$3.60\text{ per Liter}$ Better value: 5 L bottle ## Question 13 Perimeter: $44\text{ meters}$ ## Question 14 Water wasted: $1.8\text{ Liters}$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Ratios and Proportions" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 7: Calculate unit rates and compare

Question 8: Determine total volume of concrete mix

Question 9: Calculate fuel needed for 500 km

Question 10: Convert map scale to actual distance

Question 11: Calculate words typed in 7 minutes

Question 12: Calculate unit prices and compare value

Question 13: Calculate perimeter of the patio

Question 14: Calculate wasted water in liters

Answer:

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14