Solve Ratios And Proportions - TEAS Mathematics

1. To make a specialized concrete, a builder mixes cement, sand, and gravel in a ratio of 2:5:8 by weight. The sand used for the mix has a density of 1.6 g/cm³, and the builder needs to use 2.5 cubic meters of sand. If cement costs $0.80 per kilogram, what is the total cost of the cement required for the entire batch of concrete?
   Answer: $1,280 This multi-step problem requires unit conversions and proportions. Step 1: Find the weight of the sand. First, convert the volume of sand to cm³: $1$ m = $100$ cm, so $1$ m³ = $(100)^3$ cm³ = (1,000,000) cm³. The volume of sand is $2.5 \times 1,000,000 = 2,500,000$ cm³. Use the density to find the mass (weight): Mass = Density $\times$ Volume = $1.6 \text{ g/cm³} \times 2,500,000 \text{ cm³} = 4,000,000$ g. Convert this mass to kilograms: $4,000,000 \text{ g} / 1000 \text{ g/kg} = 4000$ kg. Step 2: Use the ratio to find the weight of cement. The ratio of cement to sand is 2:5. Let $C$ be the weight of cement. $\frac{C}{4000 \text{ kg}} = \frac{2}{5}$. Cross-multiply: $5C = 2 \times 4000 = 8000$. Solve for $C$: $C = 8000 / 5 = 1600$ kg. Step 3: Calculate the total cost of the cement. Cost = Weight $\times$ Price per kg = $1600 \text{ kg} \times \$0.80/\text{kg} = \$1280$.
2. A chemist has a 400 mL solution that is 15% acid by volume, with the remainder being water. How much pure acid must be added to this solution to change the ratio of acid to water to 3:4?
   Answer: 195 mL First, calculate the initial amounts of acid and water. Initial acid = 15% of 400 mL = $0.15 \times 400 = 60$ mL. Initial water = 85% of 400 mL = $0.85 \times 400 = 340$ mL. Let 'a' be the volume of pure acid to be added. When acid is added, the amount of water remains constant at 340 mL. The new amount of acid will be $60 + a$. The desired new ratio of acid to water is 3:4. Set up the proportion: $\frac{\text{new acid}}{\text{water}} = \frac{60 + a}{340} = \frac{3}{4}$. Cross-multiply: $4(60 + a) = 3(340)$. This gives $240 + 4a = 1020$. Subtract 240 from both sides: $4a = 780$. Solve for 'a': $a = 780 / 4 = 195$ mL.
3. A car and a truck start from the same point and travel in the same direction. The ratio of the car's speed to the truck's speed is 7:4. After 3 hours, the car is 72 miles ahead of the truck. How many miles will the truck have traveled in that 3-hour period?
   Answer: 96 miles Let the car's speed be $7x$ mph and the truck's speed be $4x$ mph. The difference in their speeds is $7x - 4x = 3x$ mph. This difference in speed is the rate at which the distance between them increases. The distance formula is Distance = Rate $\times$ Time. We are given that the distance between them is 72 miles after 3 hours. So, $72 \text{ miles} = (3x \text{ mph}) \times (3 \text{ hours})$. This gives $72 = 9x$. Solving for $x$ gives $x = 8$. Now, find the speed of the truck: Truck's speed = $4x = 4(8) = 32$ mph. The question asks for the distance the truck traveled in 3 hours. Distance = Speed $\times$ Time = $32 \text{ mph} \times 3 \text{ hours} = 96$ miles.
4. The scale on a blueprint is 1/4 inch : 3 feet. On the blueprint, a rectangular room measures 2.5 inches by 4 inches. A contractor determines that the actual cost of flooring for this room will be $5.25 per square yard. What is the total cost to purchase the flooring for the room?
   Answer: $840 Step 1: Convert blueprint dimensions to actual feet. The scale is 0.25 in = 3 ft. The conversion factor is $3 \text{ ft} / 0.25 \text{ in} = 12 \text{ ft/in}$. Actual length = $4 \text{ in} \times 12 \text{ ft/in} = 48$ feet. Actual width = $2.5 \text{ in} \times 12 \text{ ft/in} = 30$ feet. Step 2: Calculate the actual area in square feet. Area = Length $\times$ Width = $48 \text{ ft} \times 30 \text{ ft} = 1440$ square feet. Step 3: Convert the area to square yards. Since 1 yard = 3 feet, 1 square yard = $3^2 = 9$ square feet. Area in sq yd = $1440 \text{ sq ft} / 9 \text{ sq ft/sq yd} = 160$ square yards. Step 4: Calculate the total cost. Cost = Area in sq yd $\times$ Price per sq yd = $160 \text{ sq yd} \times \$5.25/\text{sq yd} = \$840$.
5. In a bag of marbles, the ratio of red marbles to blue marbles is initially 3:5. After 18 red marbles are added to the bag, the ratio of red to blue marbles becomes 3:2. What was the total number of marbles in the bag originally?
   Answer: 32 marbles Let the original number of red marbles be $3x$ and blue marbles be $5x$. The original total is $8x$. After adding 18 red marbles, the new number of red marbles is $3x + 18$, while the number of blue marbles remains $5x$. The new ratio is $(3x + 18) : 5x$, which is given as 3:2. Set up the proportion: $\frac{3x + 18}{5x} = \frac{3}{2}$. Cross-multiply to get $2(3x + 18) = 3(5x)$. This simplifies to $6x + 36 = 15x$. Solving for $x$ gives $36 = 9x$, so $x = 4$. The question asks for the original total number of marbles, which is $8x$. Substitute $x=4$ to get $8(4) = 32$.
6. A punch recipe calls for a ratio of $3$ parts pineapple juice to $5$ parts ginger ale. If a caterer wants to make $4.5$ gallons of punch using the recipe, how many gallons of pineapple juice are needed?
   Answer: 1.7 gallons When you encounter ratio problems, you're working with proportional relationships where parts combine to form a whole. The key is identifying what each part represents and how they relate to the total amount. This recipe uses a $3:5$ ratio of pineapple juice to ginger ale. This means for every $3$ parts pineapple juice, there are $5$ parts ginger ale, creating $3 + 5 = 8$ total parts. To find how much pineapple juice is needed in $4.5$ gallons of punch, calculate what fraction of the total is pineapple juice: $\frac{3}{8}$ of the mixture. Multiply the total volume by this fraction: $4.5 \times \frac{3}{8} = \frac{13.5}{8} = 1.6875$ gallons. Rounding to one decimal place gives $1.7$ gallons. Looking at the wrong answers: Choice B ($2.7$ gallons) would result from incorrectly using $\frac{3}{5}$ instead of $\frac{3}{8}$ - this forgets that ratios show relationships between parts, not parts to the whole. Choice C ($3.4$ gallons) comes from using $\frac{3}{4}$ as the fraction, perhaps by mistakenly thinking the ratio is $3:1$ instead of $3:5$. Choice D ($5.0$ gallons) exceeds the total volume entirely, showing a fundamental misunderstanding of the problem. Remember: in ratio problems, always add all parts together to find the denominator when converting to fractions. The ratio $a:b$ means the first quantity represents $\frac{a}{a+b}$ of the total mixture.
7. A team of 12 nurses is required to staff a hospital wing for an 8-hour shift. Due to an emergency, the wing must remain staffed for 12 hours. If 3 of the original nurses call out sick, how many additional nurses must be brought in to cover the longer shift, assuming the staffing ratio of nurses per hour of operation is maintained?
   Answer: 9 additional nurses This is a direct proportion problem. The number of nurses required is proportional to the duration of the shift. Let N be the number of nurses needed for the 12-hour shift. Set up the proportion: $\frac{12 \text{ nurses}}{8 \text{ hours}} = \frac{N \text{ nurses}}{12 \text{ hours}}$. Cross-multiply: $8N = 12 \times 12 = 144$. Solve for N: $N = 144 / 8 = 18$. So, 18 nurses are needed in total for the 12-hour shift. Originally, there were 12 nurses, but 3 called out sick, leaving $12 - 3 = 9$ nurses available. The number of additional nurses needed is the total required minus those available: $18 - 9 = 9$.
8. The ratio of Alice's savings to Bob's savings is 2:3, and the ratio of Bob's savings to Charlie's savings is 4:5. If their combined savings is a value between $150 and $200, how much money does Bob have?
   Answer: $60 First, find a common ratio for A:B:C. Given A:B = 2:3 and B:C = 4:5. To combine them, find the least common multiple of the 'B' terms (3 and 4), which is 12. Convert the ratios: A:B = 2:3 = 8:12 and B:C = 4:5 = 12:15. Thus, the combined ratio A:B:C is 8:12:15. The total number of parts is $8 + 12 + 15 = 35$. The total savings must be a multiple of 35. The multiples of 35 are 35, 70, 105, 140, 175, 210, etc. The only multiple between $150 and $200 is $175. If the total is $175, then each part is worth $\$175 / 35 = \$5$. Bob's savings corresponds to 12 parts, so Bob has $12 \times \$5 = \$60$.
9. A rectangular garden has a length-to-width ratio of 5:2. The owner increases both the length and the width by 20%. What is the ratio of the garden's new area to its original area?
   Answer: 36:25 Let the original length be $5x$ and the original width be $2x$. The original area is $A_1 = (5x)(2x) = 10x^2$. Increasing the dimensions by 20% is equivalent to multiplying them by 1.20. The new length is $1.2(5x) = 6x$ and the new width is $1.2(2x) = 2.4x$. The new area is $A_2 = (6x)(2.4x) = 14.4x^2$. The ratio of the new area to the original area is $\frac{A_2}{A_1} = \frac{14.4x^2}{10x^2} = 1.44$. To express this as a ratio of integers, write 1.44 as a fraction: $\frac{144}{100}$. Simplifying this fraction by dividing the numerator and denominator by 4 gives $\frac{36}{25}$. Therefore, the ratio is 36:25.
10. A medical pump is calibrated to deliver 2.4 liters of a saline solution over a period of 3 hours. A specific patient's prescription requires a dose of 300 milliliters. If the pump is started at 8:00:00 AM, at what precise time will the patient have received the full prescribed dose?
    Answer: 8:22:30 AM First, ensure all volume units are consistent. Convert 2.4 liters to milliliters: $2.4 \text{ L} \times 1000 \text{ mL/L} = 2400 \text{ mL}$. Now, set up a proportion to find the time (T) required to deliver 300 mL. The rate is 2400 mL per 3 hours. So, $\frac{2400 \text{ mL}}{3 \text{ hours}} = \frac{300 \text{ mL}}{T \text{ hours}}$. Cross-multiply: $2400 \times T = 3 \times 300 = 900$. Solve for T: $T = \frac{900}{2400} = \frac{9}{24} = \frac{3}{8}$ hours. To find the time in minutes and seconds, convert this fraction: $\frac{3}{8} \text{ hours} \times 60 \frac{\text{minutes}}{\text{hour}} = \frac{180}{8} = 22.5$ minutes. This is 22 minutes and 0.5 minutes. Convert the half-minute to seconds: $0.5 \text{ min} \times 60 \frac{\text{seconds}}{\text{min}} = 30$ seconds. The total time is 22 minutes and 30 seconds. Add this to the start time of 8:00:00 AM to get the end time of 8:22:30 AM.
11. To complete a project, 6 workers take 15 days, working 8 hours per day. The client moves the deadline up, so the project must be completed in just 10 days. To manage this, the company adds 3 more workers. Assuming all workers perform at the same rate, how many hours per day must they now work to meet the new deadline?
    Answer: 8 hours This problem involves an inverse proportion related to the total work required. First, calculate the total 'worker-hours' needed for the project. Total Work = (Number of workers) $\times$ (Number of days) $\times$ (Hours per day). Total Work = $6 \times 15 \times 8 = 720$ worker-hours. This total amount of work remains constant. Now, calculate the hours per day needed with the new conditions. New number of workers = $6 + 3 = 9$. New number of days = 10. Let $H$ be the new hours per day. The equation is: Total Work = (New Workers) $\times$ (New Days) $\times$ (New Hours/Day). So, $720 = 9 \times 10 \times H$. This simplifies to $720 = 90H$. Solve for $H$: $H = 720 / 90 = 8$. They must now work 8 hours per day.
12. Two variables, $p$ and $q$, are related such that $p$ is directly proportional to the square of $q$. It is known that when $p = 72$, $q = 3$. What is the value of $p$ when $q$ is increased by 50%?
    Answer: 162 The relationship is $p = kq^2$, where $k$ is the constant of proportionality. First, find $k$ using the given values: $p = 72$ when $q = 3$. So, $72 = k(3^2)$, which means $72 = 9k$. Solving for $k$ gives $k = 8$. The formula is $p = 8q^2$. Next, find the new value of $q$. The original $q$ is 3. It is increased by 50%, so the new $q$ is $3 + (0.50 \times 3) = 3 + 1.5 = 4.5$. Finally, calculate the new value of $p$ using this new $q$: $p = 8(4.5^2) = 8(20.25)$. $p = 162$.

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