Ratios and Comparisons - ISEE Upper Level Quantitative Reasoning

In a parking lot, the ratio of cars to motorcycles to bicycles is $12:3:2$ . If there are 34 total vehicles and each bicycle is replaced with a car, what will be the new ratio of cars to motorcycles?
Answer: $14:3$ When you encounter ratio problems with multiple changes, break them down step by step. First, determine the actual quantities, then apply the changes, and finally calculate the new ratio. Start by finding how many of each vehicle type exists initially. The ratio $12:3:2$ means for every 17 parts total ( $12+3+2=17$ ), there are 34 vehicles. So each part equals $34÷17=2$ vehicles. This gives us: 12 parts × 2 = 24 cars, 3 parts × 2 = 6 motorcycles, and 2 parts × 2 = 4 bicycles. Next, apply the change: each bicycle becomes a car. The 4 bicycles are replaced with 4 cars, giving us $24+4=28$ cars and still 6 motorcycles. The new ratio is $28:6$ , which simplifies to $14:3$ by dividing both terms by 2. Looking at the wrong answers: Choice A ( $7:1$ ) incorrectly reduces $14:3$ further, perhaps by mistakenly dividing by different numbers. Choice C ( $26:6$ ) likely comes from adding only 2 cars instead of 4, missing that all 4 bicycles become cars. Choice D ( $8:2$ ) appears to be a random simplification that doesn't match the actual numbers. Remember that in ratio problems involving changes, always work with actual quantities first rather than trying to manipulate the ratios directly. Convert ratios to real numbers, make the specified changes, then create your new ratio and simplify if possible.
A school's student-to-teacher ratio is $25:1$ . If the school has 450 students, and 50 students transfer out while 2 new teachers are hired, what is the new student-to-teacher ratio?
Answer: $20:1$ First find the number of teachers. With a ratio of $25:1$ and 450 students, there are $450 \div 25 = 18$ teachers. After the changes there are $450 - 50 = 400$ students and $18 + 2 = 20$ teachers, so the new ratio is $400:20 = 20:1$ . The ratio $400:18$ updates the number of students but forgets the 2 new teachers. The ratio $25:1$ assumes the ratio stays the same even though both groups changed. The ratio $19:1$ would need 380 students for 20 teachers, but 400 students remain.
In a survey, the ratio of people who prefer coffee to tea to juice is $8:5:2$ . If 45 people prefer tea, and 18 people change their preference from coffee to juice, what is the new ratio of people preferring coffee to people preferring juice?
Answer: $3:2$ Find the actual counts, apply the change, then simplify. Tea is 5 parts and equals 45 people, so one part is $45 \div 5 = 9$ people. That gives $8 \times 9 = 72$ coffee drinkers and $2 \times 9 = 18$ juice drinkers. When 18 people switch from coffee to juice, coffee becomes $72 - 18 = 54$ and juice becomes $18 + 18 = 36$ . The ratio is $54:36$ , and dividing both terms by 18 gives $3:2$ . The ratio $54:18$ subtracts the switchers from coffee but forgets to add them to juice. The ratio $4:1$ is the original $72:18$ , before anyone switched. The ratio $2:1$ results from miscounting one of the new totals. Tip: when people move between groups, one group loses exactly what the other gains; change both numbers.
A school allocates $2400 to sports and $1800 to music; what is the simplest sports:music ratio?
Answer: 4:3 This question tests ISEE Upper Level quantitative reasoning skills, specifically simplifying ratios involving larger numbers. Ratios compare quantities and should be reduced to their simplest form by dividing both terms by their greatest common divisor. The school allocates $2400 to sports and $1800 to music, creating a ratio of 2400:1800. The GCD of 2400 and 1800 is 600, so dividing both terms by 600 gives us 4:3. Choice A is correct because it represents the fully simplified ratio of sports funding to music funding. Choice B reverses the ratio, choice C shows an unsimplified version, and choice D uses incorrect values. Students should be encouraged to factor out hundreds or thousands first when working with large numbers, making it easier to find the GCD and simplify the ratio.
In a mixture, the ratio of water to orange juice is $2:3$ . If 10 liters of water and 5 liters of orange juice are added to 40 liters of this mixture, what is the new ratio of water to orange juice?
Answer: $26:29$ When you encounter ratio problems involving mixtures, the key is to track each component separately through all changes, then form the new ratio at the end. Start by finding how much water and orange juice are in the original 40-liter mixture. With a $2:3$ ratio of water to orange juice, the mixture contains $\frac{2}{2+3} = \frac{2}{5}$ water and $\frac{3}{5}$ orange juice. So there are $40 \times \frac{2}{5} = 16$ liters of water and $40 \times \frac{3}{5} = 24$ liters of orange juice initially. After adding 10 liters of water and 5 liters of orange juice, you have: - Water: $16 + 10 = 26$ liters - Orange juice: $24 + 5 = 29$ liters The new ratio is $26:29$ , which is answer choice B. Let's examine why the other answers are incorrect. Choice A ( $1:1$ ) would require equal amounts of water and orange juice, but we have 26 and 29 liters respectively. Choice C ( $3:4$ ) might tempt you if you mistakenly thought the original ratio would be preserved after adding equal proportions, but we're adding different amounts (10 vs 5 liters). Choice D ( $22:23$ ) could result from calculation errors, perhaps incorrectly determining the original mixture contents. Remember: in mixture problems, always break down the original mixture into its components first, then add the new quantities to each component separately. Don't try to work with ratios directly when adding different amounts to each part.
A recipe calls for flour and sugar in the ratio $5:2$ . If Maria uses 15 cups of flour, how much sugar should she use to maintain the same ratio?
Answer: 6 cups of sugar When you see ratio problems, you're working with proportional relationships where quantities maintain a constant relationship to each other. The key is setting up a proportion to find the unknown value. The recipe uses flour and sugar in a $5:2$ ratio, meaning for every 5 parts flour, you need 2 parts sugar. Since Maria uses 15 cups of flour, you need to find how many "parts" this represents: $15 ÷ 5 = 3$ parts. If the flour is 3 times the base amount, the sugar must also be 3 times its base amount: $2 × 3 = 6$ cups of sugar. You can also solve this using cross-multiplication: $\frac{5}{2} = \frac{15}{x}$ , which gives you $5x = 30$ , so $x = 6$ . Looking at the wrong answers: Choice B (7.5 cups) comes from incorrectly thinking you multiply the flour amount by the sugar ratio: $15 × \frac{2}{5} = 6$ , but then making an arithmetic error. Choice C (10 cups) results from confusing the ratios and thinking it's $2:1$ instead of $5:2$ . Choice D (37.5 cups) comes from mistakenly multiplying $15 × 2.5$ (which would be $15 × \frac{5}{2}$ ), completely reversing the relationship. The answer is A: 6 cups of sugar. Strategy tip: In ratio problems, always identify what one "part" equals by dividing the known quantity by its ratio number, then multiply by the unknown quantity's ratio number. This two-step approach prevents most ratio errors.
A theater budget is sets:costumes $=7:5$ ; if costumes increase $50\%$ , what is the new ratio?
Answer: 7:7.5 This question tests ISEE Upper Level quantitative reasoning skills, specifically calculating new ratios when one quantity changes by a percentage. The theater budget starts with sets:costumes = 7:5, and when costumes increase by 50%, the costume budget becomes 5 × 1.5 = 7.5, while sets remain at 7. The new ratio is therefore 7:7.5, which can also be expressed as 14:15 when both terms are multiplied by 2 to eliminate the decimal. Choice A is correct because it accurately represents the new ratio with sets at 7 and costumes at 7.5. Choice B incorrectly calculates the 50% increase, choice C reverses and miscalculates the ratio, and choice D shows the original ratio reversed. To master this concept, students should practice calculating percentage increases first, then form the new ratio, being comfortable with decimal values in ratios when appropriate.
In a photography club, the ratio of film cameras to digital cameras is $3:7$ . If there are 42 digital cameras, what is the ratio of film cameras to the total number of cameras?
Answer: $3:10$ The ratio of film to digital cameras is $3:7$ and there are 42 digital cameras, so each ratio part is $42 \div 7 = 6$ cameras. There are $3 \times 6 = 18$ film cameras. The total is $18 + 42 = 60$ cameras, so the ratio of film cameras to all cameras is $18:60$ , which simplifies to $3:10$ when both numbers are divided by 6. The ratio $7:10$ is digital cameras to the total, not film cameras. The ratios $3:7$ and $18:42$ both compare film cameras to digital cameras, which is the ratio given in the problem rather than the one asked for.
Two numbers are in the ratio $5:7$ . If the sum of the numbers is 96, what is the ratio of the smaller number to the difference between the larger and smaller numbers?
Answer: $5:2$ Write the numbers as $5x$ and $7x$ so they keep the ratio $5:7$ . Their sum is $12x = 96$ , so $x = 8$ , and the numbers are $40$ and $56$ . The difference is $56 - 40 = 16$ . The ratio of the smaller number to the difference is $40:16$ , and dividing by 8 gives $5:2$ . The ratio $40:56$ compares the smaller number with the larger number instead of with the difference. The ratio $2:5$ reverses the order, giving difference to smaller number. The ratio $5:12$ uses the sum (12 parts) in place of the difference (2 parts). Tip: in ratio-of-parts problems, the difference between $7x$ and $5x$ is $2x$ , so you can read $5:2$ straight from the parts.
A concrete mixture contains cement, sand, and gravel in the ratio $1:3:4$ . If a construction project uses 24 cubic yards of this mixture and requires an additional 6 cubic yards of sand to be added separately, what is the ratio of sand to the total volume of materials?
Answer: $1:2$ Find each amount first, then form the ratio. The ratio $1:3:4$ has $1 + 3 + 4 = 8$ parts, so the 24 cubic yards of mixture contain $\frac{3}{8} \times 24 = 9$ cubic yards of sand. Adding 6 more gives $9 + 6 = 15$ cubic yards of sand, and the total volume becomes $24 + 6 = 30$ cubic yards. The ratio of sand to total is $15:30 = 1:2$ . The ratio $15:24$ updates the sand but forgets to add the extra 6 cubic yards to the total. The ratio $9:30$ updates the total but leaves out the added sand. The ratio $3:8$ is the sand share of the original mixture, before anything was added. Tip: when material is added, update both the part and the whole before writing the ratio.
Two pipes fill a tank in the ratio $3:5$ regarding their flow rates. If the faster pipe can fill the tank alone in 15 hours, how long would it take both pipes together to fill $\frac{2}{3}$ of the tank?
Answer: 6.25 hours together When you encounter pipe flow problems, you're dealing with rates of work. The key insight is that flow rates are inversely proportional to time—faster pipes take less time to fill the same volume. Since the pipes have flow rates in the ratio $3:5$ , and the faster pipe (ratio 5) fills the tank in 15 hours, you can find the slower pipe's time. If the faster pipe has rate $\frac{1}{15}$ tanks per hour, then the slower pipe has rate $\frac{3}{5} \times \frac{1}{15} = \frac{1}{25}$ tanks per hour, meaning it takes 25 hours alone. When working together, their combined rate is $\frac{1}{15} + \frac{1}{25}$ . Finding a common denominator: $\frac{5}{75} + \frac{3}{75} = \frac{8}{75}$ tanks per hour. To fill $\frac{2}{3}$ of the tank at rate $\frac{8}{75}$ : Time = $\frac{\frac{2}{3}}{\frac{8}{75}} = \frac{2}{3} \times \frac{75}{8} = \frac{150}{24} = 6.25$ hours. Choice A (6.25 hours) is correct. Choice B (7.5 hours) likely comes from incorrectly calculating the time to fill the entire tank together, then making an error with the $\frac{2}{3}$ factor. Choice C (5.0 hours) might result from using only the faster pipe's rate for the entire calculation. Choice D (10.0 hours) could come from incorrectly finding $\frac{2}{3}$ of the slower pipe's individual time. Remember: always add the rates when pipes work together, then divide the desired volume by this combined rate.
A train travels 180 miles in 4 hours; a car travels 210 miles in 3 hours; what is train speed:car speed?
Answer: 9:14 Speed is distance divided by time. The train travels 180 ÷ 4 = 45 miles per hour, and the car travels 210 ÷ 3 = 70 miles per hour. The ratio of train speed to car speed is 45:70, which simplifies to 9:14 when both numbers are divided by 5. The ratio 6:7 compares the distances, 180:210, and ignores the different travel times. The ratio 14:9 has the right numbers in the wrong order; the question asks for the train's speed first. The ratio 3:4 compares the times, 3 hours for the car to 4 hours for the train, rather than the speeds.
Bus A travels 120 miles in 3 hours; Bus B travels 150 miles in 5 hours; what is $v_A:v_B$ ?
Answer: 4:3 Speed is distance divided by time. Bus A travels 120 ÷ 3 = 40 miles per hour, and Bus B travels 150 ÷ 5 = 30 miles per hour. The ratio of their speeds is 40:30, which simplifies to 4:3 when both numbers are divided by 10. The ratio 3:4 reverses the order; the question asks for Bus A's speed first. The ratio 4:5 compares the distances, 120:150, and ignores that the buses traveled for different amounts of time. The ratio 8:5 does not match either way of comparing the buses; the speeds 40 and 30 reduce only to 4:3.
A recipe uses 12 cups flour and 18 cups sugar for a bake sale; what is the simplest flour:sugar ratio?
Answer: 2:3 This question tests ISEE Upper Level quantitative reasoning skills, specifically simplifying ratios to their lowest terms. Ratios should be expressed in simplest form by dividing both terms by their greatest common divisor (GCD). The recipe uses 12 cups flour and 18 cups sugar, giving a ratio of 12:18. The GCD of 12 and 18 is 6, so dividing both terms by 6 gives us 2:3. Choice C is correct because it represents the fully simplified ratio of flour to sugar. Choice A shows the unsimplified ratio, choice B reverses the terms, and choice D only partially simplifies the ratio. To master this skill, students should practice finding the GCD of two numbers and understand that simplifying ratios is similar to reducing fractions to lowest terms.
A factory produces red, blue, and green widgets in the ratio $6:4:5$ . If production increases such that red widget output doubles, blue widget output increases by 50%, and green widget output remains constant, what is the new ratio of red to blue to green widgets?
Answer: $12:6:5$ Ratio problems involving changes require you to apply percentage increases to each component separately, then express the results in simplest form. Start with the original ratio of red:blue:green widgets at $6:4:5$ . Now apply each change: - Red widgets double: $6 \times 2 = 12$ - Blue widgets increase by 50%: $4 \times 1.5 = 6$ - Green widgets remain constant: $5$ This gives you the new ratio $12:6:5$ , which is already in simplest form since 12, 6, and 5 share no common factors other than 1. Looking at the wrong answers: Choice B ( $8:3:5$ ) appears to come from incorrectly calculating the blue increase as 25% instead of 50%, giving $4 \times 1.25 = 5$ , then somehow arriving at different values entirely. Choice C ( $6:2:5$ ) suggests someone misunderstood "doubles" as "increases by 100%" but applied it incorrectly, and severely miscalculated the blue increase. Choice D ( $12:8:5$ ) correctly doubles the red widgets but incorrectly doubles the blue widgets too, instead of applying the 50% increase. The correct answer is A: $12:6:5$ . Remember that "increases by X%" means you multiply by $(1 + \frac{X}{100})$ , while "doubles" means you multiply by 2. Always double-check that you're applying the right operation to each component, and verify your final ratio is in simplest form by checking for common factors.
A smoothie uses yogurt:fruit $=2:5$ ; which choice keeps the same ratio when doubled?
Answer: 4:10 This question tests ISEE Upper Level quantitative reasoning skills, specifically understanding and applying ratios to compare quantities. Ratios are a way to compare two quantities by division, showing how many times one value contains or is contained within the other. The smoothie recipe uses yogurt to fruit in a 2:5 ratio, meaning for every 2 parts yogurt, there are 5 parts fruit. When doubling a recipe, both quantities must be multiplied by the same factor to maintain the ratio, so 2:5 becomes 4:10. Choice C is correct because 4:10 represents the doubled quantities while maintaining the same proportional relationship as 2:5. Choice A (4:12) incorrectly doubles the first term but more than doubles the second, while choices B and D don't maintain the original ratio structure. To master this skill, students should practice scaling ratios by multiplying both terms by the same factor and verify their work by simplifying the result back to the original ratio.
A recipe uses oil:vinegar $=5:2$ ; which choice maintains the same ratio when scaled up?
Answer: 15:6 This question tests ISEE Upper Level quantitative reasoning skills, specifically recognizing when ratios are equivalent through scaling. Ratios maintain their proportional relationship when both terms are multiplied by the same factor. The recipe uses oil:vinegar = 5:2, and to scale this up, we multiply both terms by the same number. Multiplying by 3 gives us 15:6, which maintains the same proportional relationship as 5:2. Choice B is correct because 15:6 represents the original ratio scaled up by a factor of 3. Choice A (10:5) reverses the ratio, choice C (7:4) uses unrelated values, and choice D (2:5) reverses the original ratio. Students should practice scaling ratios by choosing a multiplier and applying it to both terms, then verifying by simplifying back to the original ratio.
In a chess tournament, the ratio of wins to losses to draws for Player A is $4:1:2$ . Player A has played 21 games total. If Player A wins the next 3 games, what will be the new ratio of wins to total games played?
Answer: $5:8$ The ratio $4:1:2$ has $4 + 1 + 2 = 7$ parts. With 21 games, each part is $21 \div 7 = 3$ games, so Player A has $4 \times 3 = 12$ wins, $1 \times 3 = 3$ losses, and $2 \times 3 = 6$ draws. After winning the next 3 games, Player A has $12 + 3 = 15$ wins and $21 + 3 = 24$ games played. The ratio of wins to total games is $15:24$ , which simplifies to $5:8$ by dividing both terms by 3. The ratio $15:21$ adds the 3 new wins but forgets that the total number of games also grows by 3. The ratios $12:21$ and $4:7$ describe the record before the extra games, so they ignore the change the question asks about. The new ratio is $5:8$ .
The ratio of red marbles to blue marbles in a jar is $7:3$ . If the total number of marbles is 150, and 20 green marbles are added to the jar, what will be the new ratio of red marbles to the total number of marbles?
Answer: $21:34$ Split the 150 marbles in the ratio $7:3$ : there are 10 parts of 15 marbles each, so there are $7 \times 15 = 105$ red and $3 \times 15 = 45$ blue marbles. Adding 20 green marbles makes the total $150 + 20 = 170$ . The ratio of red marbles to all marbles is $105:170$ ; dividing both numbers by 5 gives $21:34$ . The ratio $105:150$ forgets to add the 20 green marbles to the total. The ratios $7:17$ and $7:13$ keep the 7 from the original ratio and add whole ratio parts for the green marbles, but 20 green marbles are $20 \div 15 = 1\frac{1}{3}$ parts, so the new total cannot be written that way; work with actual counts instead.
Two machines produce widgets in the ratio $5:3$ . If Machine A produces 15 widgets per hour, what is the ratio of Machine B's hourly production to the total hourly production of both machines?
Answer: $3:8$ Machine A makes 15 widgets per hour and the ratio A:B is $5:3$ , so each ratio part is $15 \div 5 = 3$ widgets. Machine B makes $3 \times 3 = 9$ widgets per hour, and together the machines make $15 + 9 = 24$ widgets per hour. The ratio of Machine B's production to the total is $9:24$ , which simplifies to $3:8$ when both numbers are divided by 3. The ratio $5:8$ is Machine A's share of the total, not Machine B's. The ratio $9:15$ compares Machine B to Machine A instead of to the total. The ratio $1:3$ would mean Machine B makes one third of the total, but 9 out of 24 is three eighths, which is more than one third.

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