Ratios and Rates in Context - ISEE Middle Level Quantitative Reasoning

1. Which ratio matches the statement: "$8$ girls and $12$ boys" as girls to boys?

   $8:12$ (equivalently $2:3$). The ratio of girls to boys is directly $8:12$, which simplifies by dividing by $4$ to $2:3$.

2. Identify the unit rate for $15$ dollars for $6$ pounds of apples.

   $2.50 per pound. Divide total cost by pounds to compute unit rate as $15 \div 6 = 2.5$ dollars per pound.

3. What is the ratio of $30$ minutes to $2$ hours in simplest form?

   $30:120 = 1:4$. Convert $2$ hours to $120$ minutes, then simplify $30:120$ by dividing by $30$ to $1:4$.

4. What is the speed in feet per second for $300$ feet in $10$ seconds?

   $30$ feet per second. Divide distance by time to find the rate as $300 \div 10 = 30$ feet per second.

5. Find the cost at a unit rate of $4 per ticket for $7$ tickets.

   $28. Multiply the unit rate by the number of tickets as $\ 4 \times 7 = \$28$ to find total cost.

6. Find the unit rate in miles per gallon if a car travels $180$ miles on $6$ gallons.

   $30$ miles per gallon. Divide miles by gallons to find unit rate as $180 \div 6 = 30$ miles per gallon.

7. Find the number of cups of water needed if the ratio water:rice is $2:1$ and there are $3$ cups of rice.

   $6$ cups of water. Using the ratio $2:1$, scale by $3$ for rice to find water as $2 \times 3 = 6$ cups.

8. What does the ratio $5:2$ mean when comparing cats to dogs in a shelter?

   There are $5$ cats for every $2$ dogs. The ratio $5:2$ indicates the comparative number of cats to dogs present.

9. What is the meaning of the ratio $\frac{2}{5}$ when comparing wins to total games?

   $2$ wins out of every $5$ games. The ratio $\frac{2}{5}$ expresses wins as a fraction of total games played.

10. What is the unit price if $5$ notebooks cost $12.50?

    $2.50 per notebook. Divide total cost by number of notebooks to find the price per unit as $\frac{\$12.50}{5} = \$2.50$.

11. What is the time needed at a rate of $12$ miles per hour to travel $36$ miles?

    $3$ hours. Divide distance by rate as $36 \div 12 = 3$ to determine the time required.

12. What is the meaning of the ratio $\frac{7}{2}$ in the context of miles per hour?

    $7$ miles for every $2$ hours. The ratio $\frac{7}{2}$ represents a rate of $7$ miles traveled in $2$ hours.

13. What is the unit rate for $18$ pages in $3$ minutes?

    $6$ pages per minute. Divide total pages by minutes to compute the unit rate as $18 \div 3 = 6$.

14. What is the meaning of "per" in a rate such as $9$ dollars per hour?

    For each $1$ hour, the amount is $9. The word 'per' in a rate indicates the quantity associated with each single unit of the denominator.

15. What is the ratio of boys to total students if there are $12$ boys and $8$ girls?

    $12:20$ (equivalently $3:5$). Boys to total students is $12:(12+8)=12:20$, simplified by dividing by $4$ to $3:5$.

16. What is the ratio of shaded to total if $9$ of $12$ squares are shaded, simplified?

    $9:12 = 3:4$. The ratio $9:12$ simplifies to $3:4$ by dividing both by $3$ for shaded to total squares.

17. What does the ratio $3:5$ mean when comparing apples to oranges?

    There are $3$ apples for every $5$ oranges. The ratio $3:5$ compares the number of apples to oranges, indicating $3$ apples per $5$ oranges.

18. Identify the better buy: $12$ ounces for $3 or $20$ ounces for $4.

    $20$ ounces for $4 (lower $\$/$ ounce). Compare unit prices: $\frac{\$3}{12} = \$0.25$ vs. $\frac{\$4}{20} = \$0.20$, so the lower rate is better.

19. Identify the equivalent ratio to $6:9$ in simplest form.

    $2:3$. Simplify $6:9$ by dividing both terms by $3$ to obtain the equivalent ratio $2:3$.

20. What does a ratio of $1:4$ mean in a paint mixture of red to white?

    $1$ part red for every $4$ parts white. The ratio $1:4$ denotes the proportion of red paint to white paint in the mixture.

21. What is the meaning of the rate $3 per pound in a grocery context?

    Each $1$ pound costs $3. The rate $\frac{\$3}{1}$ pound specifies the cost for each single pound of an item.

22. Find the distance at a rate of $55$ miles per hour for $2$ hours.

    $110$ miles. Multiply rate by time as $55 \times 2 = 110$ to calculate total distance traveled.

23. What is the meaning of a constant rate when a situation is proportional?

    The ratio $\frac{y}{x}$ stays the same for all pairs. In proportional situations, a constant rate means the ratio of output to input remains unchanged.

24. Which statement correctly interprets the ratio $4:1$ as students to teachers?

    There are $4$ students for every $1$ teacher. The ratio $4:1$ compares students to teachers, meaning $4$ students per teacher.

25. Identify the rate described by "$240$ miles in $4$ hours" in miles per hour.

    $60$ miles per hour. Divide distance by time to express the rate as $240 \div 4 = 60$ miles per hour.

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