PSAT Math Flashcards: Rates

What this deck covers

This deck focuses on Rates, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the time for combined work: Using the machines from the previous question, how many minutes per part together?

Answer: $2\text{ minutes per part}$. Reciprocal of combined rate: $\frac{1}{\frac{1}{2}} = 2$.

Flashcard 2: A machine makes $120$ parts in $8$ hours. What is the rate in parts per hour?

Answer: $15$ parts per hour. Divide total parts by hours: rac{120}{8}=15.

Flashcard 3: What time is needed to travel $150$ miles at $50\text{ mph}$?

Answer: $3\text{ hours}$. Using $t = \frac{d}{r}$: $150 \div 50 = 3$ hours.

Flashcard 4: What is the conversion from $\text{mi/h}$ to $\text{ft/s}$ for a speed of $1\ \text{mi/h}$?

Answer: $1\ \text{mi/h}=\frac{22}{15}\ \text{ft/s}$. Use $\frac{5280\text{ ft}}{3600\text{ s}}=\frac{22}{15}\text{ ft/s}$.

Flashcard 5: What is the slope (rate) of the line through $(2,3)$ and $(6,11)$?

Answer: $2$. Use $m=\frac{y_2-y_1}{x_2-x_1}$: $\frac{11-3}{6-2}=\frac{8}{4}=2$.

Flashcard 6: If a pipe fills a tank in $6$ hours, what fraction of the tank is filled per hour?

Answer: rac{1}{6} of the tank per hour. Rate is reciprocal of time to complete one job.

Flashcard 7: What is the slope-intercept form of a line, showing constant rate of change?

Answer: $y=mx+b$. Standard form where $m$ is the rate and $b$ is initial value.

Flashcard 8: What time is needed to travel $90$ miles at $45\ \text{mi/h}$?

Answer: $2\ \text{hours}$. Divide distance by rate: $\frac{90}{45}=2$.

Flashcard 9: What is the percent rate formula for $p\%$ written as a decimal?

Answer: $\frac{p}{100}$. Convert percent to decimal by dividing by 100.

Flashcard 10: What distance is traveled at $45\text{ mi/hr}$ for $2$ hours?

Answer: $90\text{ mi}$. Apply $d = rt = 45 \times 2$.

Flashcard 11: What is the percent increase from $50$ to $65$?

Answer: $30\%$. $\frac{65-50}{50} \times 100\% = 30\%$ increase.

Flashcard 12: If $y=kx$ with $k=2.5$, what is $y$ when $x=12$?

Answer: $y=30$. Substitute into direct variation: $y=2.5 imes 12=30$.

Flashcard 13: What is the density if mass is $36$ g and volume is $12$ cm$^3$?

Answer: $3$ g per cm$^3$. $\rho = \frac{36}{12} = 3$ g/cm³.

Flashcard 14: What is the conversion factor to change hours to minutes?

Answer: $1\text{ hr}=60\text{ min}$. Standard time conversion for hour-to-minute calculations.

Flashcard 15: State the formula for distance traveled at constant speed.

Answer: $d = rt$. Distance equals rate times time for constant motion.

Flashcard 16: State the formula for distance in terms of speed and time.

Answer: $ext{distance}= ext{speed} imes ext{time}$. Rearranged speed formula by multiplying both sides by time.

Flashcard 17: What is the formula for distance in terms of speed $v$ and time $t$?

Answer: $d=vt$. Multiply speed by time to find distance.

Flashcard 18: What is the unit rate if $6$ pounds of apples cost $15?

Answer: $2.50 per pound. Divide total cost by quantity: rac{15}{6}=2.50.

Flashcard 19: What is the combined rate if machine A makes $1$ item in $6$ min and B makes $1$ item in $3$ min?

Answer: $\frac{1}{2}$ item per minute. Add individual rates: $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ per minute.

Flashcard 20: What is the combined work rate formula for two workers with rates $r_1$ and $r_2$?

Answer: $r_{\text{total}}=r_1+r_2$. Add individual rates when working together.

Flashcard 21: What is the reciprocal rate of $a$ miles per hour?

Answer: rac{1}{a} hours per mile. Flip the fraction to get reciprocal rate.

Flashcard 22: State the formula for unit price as a rate.

Answer: ext{unit price}=rac{ ext{cost}}{ ext{items}}. Total cost divided by quantity gives cost per item.

Flashcard 23: What is the formula for price per unit if total cost is $C$ for $n$ items, with $n \ne 0$?

Answer: $\frac{C}{n}$ dollars per item. Unit price equals total cost divided by quantity.

Flashcard 24: Convert $3\text{ m/s}$ to $\text{m/min}$.

Answer: $180\text{ m/min}$. Multiply by $\frac{60\text{ s}}{1\text{ min}}$ to convert units.

Flashcard 25: What is the constant of proportionality $k$ in the direct variation equation $y=kx$ if $y=5x$?

Answer: $k=5$. In $y=kx$, the coefficient of $x$ is $k$.

Flashcard 26: What is the formula for density in terms of mass $m$ and volume $V$?

Answer: $\rho=\frac{m}{V}$. Density equals mass divided by volume.

Flashcard 27: A printer makes $180$ pages in $6$ minutes. What is the rate in $\text{pages/min}$?

Answer: $30\text{ pages/min}$. Divide total pages by time: $\frac{180}{6}$.

Flashcard 28: What is the time needed to travel $90$ miles at $45$ miles per hour?

Answer: $2$ hours. $t = \frac{d}{r} = \frac{90}{45} = 2$ hours.

Flashcard 29: What conversion factor changes $\text{hours}$ to $\text{minutes}$ in a rate problem?

Answer: $1\text{ hour} = 60\text{ minutes}$. Standard time conversion for rate calculations.

Flashcard 30: Find the slope (rate of change) between $(2,5)$ and $(6,17)$.

Answer: $3$. Use slope formula: rac{17-5}{6-2}=rac{12}{4}=3.

Flashcard 31: What is the relationship between rate $r$, time $t$, and work $W$ for constant productivity?

Answer: $W=rt$. Work equals rate times time for constant productivity.

Flashcard 32: Identify the unit rate: A car travels $150$ miles in $3$ hours. What is the speed?

Answer: $50$ miles per hour. Divide total distance by total time: rac{150}{3}=50.

Flashcard 33: What is the unit price if $5$ notebooks cost $12.50$?

Answer: $2.50$ per notebook. Divide total cost by number of items: rac{12.50}{5}=2.50.

Flashcard 34: State the direct variation equation relating $y$, $k$, and $x$.

Answer: $y=kx$. Shows $y$ is a constant multiple of $x$.

Flashcard 35: What is the time formula for one job when the combined work rate is $r$ jobs per hour?

Answer: $t=\frac{1}{r}$ hours. Time equals 1 divided by the rate.

Flashcard 36: A car goes $60$ miles in $1$ hr, then $60$ miles in $2$ hr. What is the average speed?

Answer: $40\text{ mi/hr}$. Average speed: $\frac{\text{total distance}}{\text{total time}} = \frac{120}{3}$.

Flashcard 37: Identify the unit price: $12 for $8$ pounds equals how many dollars per pound?

Answer: \1.50\text{ per lb}$.$12 \div 8 = 1.50$ dollars per pound.

Flashcard 38: How long to make $1$ item together if A makes $1$ item in $6$ min and B makes $1$ item in $3$ min?

Answer: $2\text{ minutes}$. Reciprocal of combined rate: $\frac{1}{\frac{1}{2}} = 2$ minutes.

Flashcard 39: What is the conversion from $1$ mile to feet?

Answer: $1\text{ mi} = 5280\text{ ft}$. Standard distance conversion in US customary units.

Flashcard 40: What is the combined work rate if one person works at $r_1$ job per hour and another at $r_2$ job per hour?

Answer: $r_1+r_2$ job per hour. Add individual rates when working simultaneously.

Flashcard 41: What is the relative speed when two objects move in the same direction at $v_1>v_2$?

Answer: $v_1-v_2$. When moving same direction, subtract slower from faster speed.

Flashcard 42: Two people start $120$ miles apart and drive toward each other at $50$ and $30\text{ mi/hr}$. What is the closing speed?

Answer: $80\text{ mi/hr}$. Add speeds when moving toward each other: $50+30=80$.

Flashcard 43: Identify the average speed: travel $60$ miles in $1$ hour, then $60$ miles in $2$ hours.

Answer: $40$ miles per hour. $\frac{120}{3} = 40$ mph for total distance over total time.

Flashcard 44: What is the formula for time in terms of distance $d$ and speed $v$?

Answer: $t=\frac{d}{v}$. Divide distance by speed to find time.

Flashcard 45: How many hours for two workers together if their combined work rate is $\frac{1}{4}$ job per hour?

Answer: $4$ hours. $t = \frac{1}{\frac{1}{4}} = 4$ hours.

Flashcard 46: State the formula for distance in terms of speed and time.

Answer: $ext{distance}= ext{speed} imes ext{time}$. Rearranges speed formula by multiplying both sides by time.

Flashcard 47: Compute a percent increase: A price rises from $50$ to $60$. What is the percent increase?

Answer: $20\%$. Use rac{60-50}{50} imes 100%=rac{10}{50} imes 100%.

Flashcard 48: Identify the unit rate: $18 for $6$ tickets.

Answer: $3 per ticket. Divide total cost by number of tickets: \frac{\18}{6}$.

Flashcard 49: What is the formula for speed in terms of distance $d$ and time $t$?

Answer: $v=\frac{d}{t}$. Speed equals distance divided by time.

Flashcard 50: State the formula for speed in terms of distance and time.

Answer: ext{speed}=rac{ ext{distance}}{ ext{time}}. Fundamental formula relating motion quantities.

Flashcard 51: What is the better buy per ounce: $12$ oz for $3 or $20$ oz for $4?

Answer: $20$ oz for $4. Compare $\frac{3}{12} = 0.25$ vs $\frac{4}{20} = 0.20$ per oz.

Flashcard 52: What is the constant of proportionality $k$ if $y$ varies directly with $x$?

Answer: k=rac{y}{x}. The constant ratio between proportional quantities.

Flashcard 53: What is the formula for percent change from original $A$ to new $B$?

Answer: $\frac{B-A}{A}\times 100\%$. Subtract original from new, divide by original, multiply by $100\%$.

Flashcard 54: What distance is traveled in $2.5$ hours at $40$ miles per hour?

Answer: $100$ miles. $d = 40 \times 2.5 = 100$ miles.

Flashcard 55: State the formula for a constant rate: distance in terms of rate and time.

Answer: $d=rt$. Distance equals rate multiplied by time.

Flashcard 56: State the formula for density.

Answer: $\text{density}=\frac{\text{mass}}{\text{volume}}$. Mass per unit volume measures how compact matter is.

Flashcard 57: What is the formula for average rate of change from $(x_1,y_1)$ to $(x_2,y_2)$ with $x_2 \ne x_1$?

Answer: $\frac{y_2-y_1}{x_2-x_1}$. Change in $y$ divided by change in $x$ gives the slope.

Flashcard 58: State the formula for speed as a rate.

Answer: ext{speed}=rac{ ext{distance}}{ ext{time}}. Fundamental rate formula relating motion quantities.

Flashcard 59: Convert $72\text{ mi/hr}$ to $\text{ft/s}$.

Answer: $105.6\text{ ft/s}$. Multiply by $\frac{5280\text{ ft}}{1\text{ mi}}\times\frac{1\text{ hr}}{3600\text{ s}}$.

Flashcard 60: What is the speed if you travel $45$ miles in $\frac{3}{4}$ hour?

Answer: $60$ miles per hour. Using $r = \frac{d}{t}$: $\frac{45}{\frac{3}{4}} = 45 \times \frac{4}{3} = 60$ mph.

Flashcard 61: Find the distance: A cyclist rides at $12$ miles per hour for $2.5$ hours. How far?

Answer: $30$ miles. Use $d=st$: $12 imes 2.5=30$ miles.

Flashcard 62: State the formula for speed in terms of distance and time.

Answer: ext{speed}=rac{ ext{distance}}{ ext{time}}. Fundamental formula relating motion quantities.

Flashcard 63: What is the average rate of change of $y$ with respect to $x$ from $x=a$ to $x=b$?

Answer: $\frac{y(b)-y(a)}{b-a}$. Measures how much $y$ changes per unit change in $x$.

Flashcard 64: What is the cost of $7$ pounds at $2.40 per pound?

Answer: $16.80. $7 \times 2.40 = 16.80$ dollars.

Flashcard 65: What is the definition of unit rate, and how is it written as a fraction?

Answer: A rate per $1$ unit, written as $\frac{a}{1}$ or $a$ per $1$. Unit rates show the amount per one unit of the denominator.

Flashcard 66: What is the combined work rate if one machine does $\frac{1}{6}$ job/hr and another does $\frac{1}{3}$ job/hr?

Answer: $\frac{1}{2}$ job per hour. $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$.

Flashcard 67: Find the time to travel $90$ miles at $45\text{ mi/hr}$.

Answer: $2\text{ hr}$. Rearrange $d=rt$ to get $t=\frac{d}{r}=\frac{90}{45}=2$.

Flashcard 68: State the combined work formula for two workers with times $a$ and $b$ (hours per job).

Answer: rac{1}{a}+rac{1}{b}=rac{1}{t}. Combined rates add when working together.

Flashcard 69: If one worker finishes a job in $8$ hours and another in $12$ hours, what is $t$ together?

Answer: t=rac{24}{5} hours. Use rac{1}{8}+rac{1}{12}=rac{1}{t}; solve for $t$.

Flashcard 70: Identify the unit rate: $18 for $6$ pounds of apples. What is the cost per pound?

Answer: $3 per pound. $\frac{18}{6} = 3$ dollars per pound.

Flashcard 71: State the formula for distance when traveling at constant speed for a given time.

Answer: $d = rt$. Distance equals rate times time for constant speed motion.

Flashcard 72: Find the distance: Traveling at $60\text{ mph}$ for $2.5$ hours, what distance is covered?

Answer: $150\text{ miles}$. Apply $d = rt$: $60 \times 2.5 = 150$.

Flashcard 73: What is the constant rate of change if a value increases from $20$ to $35$ in $5$ hours?

Answer: $3$ per hour. $\frac{35-20}{5} = \frac{15}{5} = 3$ per hour.

Flashcard 74: What is the formula for speed when distance is $d$ and time is $t$ (with $t \ne 0$)?

Answer: $r = \frac{d}{t}$. Rate equals distance divided by time.

Flashcard 75: What is the time to finish $1$ job at constant work rate $r$ jobs per hour, with $r \ne 0$?

Answer: $\frac{1}{r}$ hours. Time is reciprocal of rate for one job.

Flashcard 76: What is the definition of a rate in math?

Answer: A ratio comparing two quantities with different units. Rates express relationships between different measurement units.

Flashcard 77: Identify the unit rate: $180$ miles in $3$ hours equals how many miles per hour?

Answer: $60$ miles per hour. $\frac{180}{3} = 60$ miles per hour.

Flashcard 78: Identify the unit price: $12$ apples cost $6. What is the cost per apple?

Answer: $0.50 per apple. Divide total cost by quantity: rac{\6}{12}=$0.50$.

Flashcard 79: Find the unit price: $12 for $8$ notebooks. What is the cost per notebook?

Answer: \1.50\text{ per notebook}$. Divide total cost by quantity:$\frac{12}{8} = 1.50$.

Flashcard 80: What is the definition of a rate?

Answer: A ratio comparing two quantities with different units. Rates express relationships between different measurement types.

Flashcard 81: What is the unit price for $6$ items costing $18$ dollars?

Answer: $3$ dollars per item. Divide $18 by $6$ items to get price per item.

Flashcard 82: If $y$ varies directly with $x$ and $y=18$ when $x=6$, what is $k$?

Answer: $k=3$. Use k=rac{y}{x}: k=rac{18}{6}=3.

Flashcard 83: What is the combined work time if one worker takes $6$ hours and another takes $3$ hours?

Answer: $2$ hours. Combined rate $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$, so time is $2$ hours.

Flashcard 84: Find the constant of proportionality: If $y=18$ when $x=6$ in a proportional relationship, what is $k$?

Answer: $3$. Find $k$ using k=rac{y}{x}=rac{18}{6}=3.

Flashcard 85: A printer makes $120$ pages in $3$ minutes. How many pages per minute is this rate?

Answer: $40$ pages per minute. $\frac{120}{3} = 40$ pages per minute.

Flashcard 86: What is the formula for distance in terms of speed $v$ and time $t$?

Answer: $d=vt$. Rearrange $v=\frac{d}{t}$ to solve for distance.

Flashcard 87: What distance is traveled at $60\ \text{mi/h}$ for $2.5$ hours?

Answer: $150\ \text{miles}$. Multiply rate by time: $60 \times 2.5=150$.

Flashcard 88: What is the formula for time when $d$ is distance and $r$ is rate?

Answer: $t=\frac{d}{r}$. Time equals distance divided by rate.

Flashcard 89: What is the formula for a population after $t$ hours with initial $P_0$ and constant rate $r$ per hour?

Answer: $P=P_0+rt$. Linear growth: initial value plus rate times time.

Flashcard 90: What is the formula for rate when $d$ is distance and $t$ is time?

Answer: $r=\frac{d}{t}$. Rate equals distance divided by time.

Flashcard 91: Identify the equation that represents a proportional relationship with unit rate $k$.

Answer: $y = kx$. Direct variation means $y$ is proportional to $x$.

Flashcard 92: Two pipes fill a tank in $6$ hours and $3$ hours. How long to fill $1$ tank together?

Answer: $2$ hours. $\frac{1}{\frac{1}{2}} = 2$ hours to fill one tank.

Flashcard 93: What is the formula for constant speed in terms of distance $d$ and time $t$?

Answer: $v=\frac{d}{t}$. Speed equals distance divided by time.

Flashcard 94: What is the unit rate if a car travels $180$ miles in $3$ hours?

Answer: $60$ miles per hour. Divide total distance by total time: $\frac{180}{3}=60$.

Flashcard 95: What is the time to finish $1$ job with combined work rate $r_1+r_2$ job per hour?

Answer: $\frac{1}{r_1+r_2}$ hours. Time equals work divided by combined rate.

Flashcard 96: What is the percent decrease from $80$ to $60$?

Answer: $25\%$. Use $\frac{60-80}{80}\times 100\%=\frac{-20}{80}\times 100\%=25\%$.

Flashcard 97: State the formula for speed when distance traveled and time are known.

Answer: $r = \frac{d}{t}$. Rate equals distance divided by time.

Flashcard 98: Two pipes fill a tank in $6$ hours and $3$ hours. What is the combined fill rate in tanks per hour?

Answer: $\frac{1}{2}$ tank per hour. $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$.

Flashcard 99: One pipe fills a tank in $6$ hours. What is its fill rate in tanks per hour?

Answer: $\frac{1}{6}$ tank per hour. Rate is reciprocal of time: $\frac{1}{6}$ tank per hour.

Flashcard 100: What is the speed in $\text{mi/hr}$ if you travel $150$ miles in $3$ hours?

Answer: $50\text{ mi/hr}$. Apply $r = \frac{d}{t} = \frac{150}{3}$.