ACT Math Flashcards: Word Problems

What this deck covers

This deck focuses on Word Problems, giving you a quick way to review the definitions, rules, and examples that matter most for ACT 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: A $10$ L solution is $30\%$ acid; how many liters of acid does it contain?

Answer: $3$ L. $10 \times 0.30 = 3$ liters of acid

Flashcard 2: A price increases from $50$ to $60$ dollars; what is the percent increase?

Answer: $20\%$. $\frac{60-50}{50} \times 100\% = 20\%$

Flashcard 3: How do you set up a system of equations for a word problem?

Answer: Translate conditions into algebraic equations. Convert word relationships into mathematical expressions.

Flashcard 4: Identify the translation: "twice a number $x$" becomes what expression?

Answer: $2x$. Multiply the number $x$ by 2.

Flashcard 5: What does the phrase "of" usually indicate in an ACT word problem expression?

Answer: "Of" usually means multiplication, as in $\frac{3}{4}$ of $x$ is $\frac{3}{4}x$. This indicates multiplication in word problems.

Flashcard 6: What is the total amount formula for simple interest with principal $P$ and interest $I$?

Answer: $A=P+I$. Total amount is principal plus accumulated interest.

Flashcard 7: A price decreases from $80$ to $68$ dollars; what is the percent decrease?

Answer: $15\%$. $\frac{80-68}{80} \times 100\% = 15\%$

Flashcard 8: A taxi fare is $3 plus $0.50 per mile; if the fare is $8, how many miles were traveled?

Answer: $10$ miles. $8 = 3 + 0.50m$, so $m = 10$ miles.

Flashcard 9: A taxi fare is $3 plus $0.50 per mile; if the fare is $8, how many miles were traveled?

Answer: $10$ miles. $8 = 3 + 0.50m$, so $m = 10$ miles.

Flashcard 10: State the formula to calculate distance in word problems.

Answer: Distance = Speed × Time. Fundamental motion formula relating three key variables.

Flashcard 11: What is the first step in solving ratio problems?

Answer: Set up a proportion. Creates equivalent fractions to solve for unknowns.

Flashcard 12: How do you solve a percent decrease word problem?

Answer: New Value = Original Value × (1 - Percent Decrease). Multiplies by $(1 - \text{decimal percent})$ for decrease.

Flashcard 13: What is the key formula for solving average problems?

Answer: Average = Total Sum / Number of Items. Divides total by count to find mean value.

Flashcard 14: Find the original price if an item costs $36$ after a $10\%$ discount.

Answer: $40$. $36 = 0.90x$, so $x = 40$

Flashcard 15: How do you express 'half of a number' in algebraic terms?

Answer: $\frac{x}{2}$. Division by 2 represents taking half the value.

Flashcard 16: What equation represents 'Three times a number decreased by 4 equals 11'?

Answer: Equation: $3x - 4 = 11$. Translates 'three times' as $3x$ and 'decreased by 4' as $-4$.

Flashcard 17: Identify the translation: "a number $x$ less than $7$" becomes what expression?

Answer: $7-x$. Subtract $x$ from 7, not 7 from $x$.

Flashcard 18: How do you write 'the product of 5 and a number is 45' as an equation?

Answer: 5x = 45. Product relationship between 5 and unknown variable.

Flashcard 19: State the formula to calculate distance in word problems.

Answer: Distance = Speed × Time. Fundamental motion formula relating three key variables.

Flashcard 20: What is the equation for 'A number is divided by 4 to give 3'?

Answer: Equation: $\frac{x}{4} = 3$. Translates division phrase into fractional equation.

Flashcard 21: What is the key step in solving rate problems involving two objects?

Answer: Set equations based on relative rates. Compare speeds to determine relative motion.

Flashcard 22: A jar has $5$ red and $3$ blue marbles; what is the probability of drawing a red marble?

Answer: $\frac{5}{8}$. 5 red out of 8 total marbles.

Flashcard 23: What is the first step in solving a work-rate word problem?

Answer: Determine the rate of work for each individual or machine. Individual rates combine to find total completion time.

Flashcard 24: How do you express 'half of a number' in algebraic terms?

Answer: $\frac{x}{2}$. Division by 2 represents taking half the value.

Flashcard 25: Worker A completes $\frac{1}{4}$ of a job per hour; how many hours for the whole job?

Answer: $4$ hours. Rate is $\frac{1}{4}$ job per hour, so 4 hours total.

Flashcard 26: A $10$ L solution is $30\%$ acid; how many liters of acid does it contain?

Answer: $3$ L. $10 \times 0.30 = 3$ liters of acid

Flashcard 27: How do you express consecutive even numbers algebraically?

Answer: Use $x, x+2, x+4, \text{etc.}$. Even numbers differ by 2, starting from any even $x$.

Flashcard 28: How do you express consecutive odd numbers algebraically?

Answer: Use $x, x+2, x+4, \text{etc.}$. Odd numbers differ by 2, starting from any odd $x$.

Flashcard 29: What is the formula to find the distance in a distance-speed-time problem?

Answer: Distance = Speed × Time. Fundamental relationship connecting these three quantities.

Flashcard 30: How do you express consecutive even numbers algebraically?

Answer: Use $x, x+2, x+4, \text{etc.}$. Even numbers differ by 2, starting from any even $x$.

Flashcard 31: How do you express 'a number divided by 5' in algebraic terms?

Answer: $\frac{x}{5}$. Division operation represented as a fraction.

Flashcard 32: What is the equation for a simple interest word problem?

Answer: Interest = Principal × Rate × Time. Standard formula for calculating simple interest earned.

Flashcard 33: A store charges $5$ fixed plus $2$ per item; what is the cost for $7$ items?

Answer: $19$. $5 + 7(2) = 19$ dollars total.

Flashcard 34: How is 'twice the sum of a number and 4' expressed algebraically?

Answer: 2(x + 4). Multiply 2 by the entire sum in parentheses.

Flashcard 35: A right triangle has legs $6$ and $8$; what is the hypotenuse length?

Answer: $10$. Use Pythagorean theorem: $\sqrt{6^2 + 8^2} = 10$.

Flashcard 36: Identify the translation: "$5$ more than a number $x$" becomes what expression?

Answer: $x+5$. Add 5 to the number $x$.

Flashcard 37: Identify the translation: "$7$ less than a number" means what expression in $x$?

Answer: $x-7$. Subtract 7 from the number $x$.

Flashcard 38: How do you express 'a number divided by 5' in algebraic terms?

Answer: $\frac{x}{5}$. Division operation represented as a fraction.

Flashcard 39: What is $30\%$ of $80$?

Answer: $24$. $0.30 \times 80 = 24$

Flashcard 40: How do you write 'the product of 5 and a number is 45' as an equation?

Answer: 5x = 45. Product relationship between 5 and unknown variable.

Flashcard 41: A job takes A $6$ hours alone and B $3$ hours alone; how long together at constant rates?

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

Flashcard 42: What is the total amount formula for simple interest with principal $P$ and interest $I$?

Answer: $A=P+I$. Total amount is principal plus accumulated interest.

Flashcard 43: What equation represents 'a number minus 6 equals 14'?

Answer: x - 6 = 14. Subtraction equation with variable and constant.

Flashcard 44: What equation represents 'the product of 6 and a number is 36'?

Answer: 6x = 36. Multiplication equation with variable and constant.

Flashcard 45: Identify the equation for 'The sum of a number and 9 is 15.'

Answer: Equation: $x + 9 = 15$. Direct translation of 'sum' as addition operation.

Flashcard 46: What is the weighted average concentration formula for mixing solutions?

Answer: $\frac{c_1a+c_2b}{a+b}$. Weighted average using concentration and amounts.

Flashcard 47: What equation represents 'Four more than twice a number is 10'?

Answer: Equation: $2x + 4 = 10$. Translates 'twice a number' as $2x$ and 'four more' as $+4$.

Flashcard 48: What is the equation for 'the product of a number and 4 is 20'?

Answer: 4x = 20. Multiplication relationship between variable and constant.

Flashcard 49: How do you calculate the selling price with a discount?

Answer: Selling Price = Original Price × (1 - Discount Rate). Multiplies by $(1 - \text{decimal discount})$ to reduce price.

Flashcard 50: State the formula for calculating the average in word problems.

Answer: Average = $\frac{\text{Sum of values}}{\text{Number of values}}$. Divides total sum by count of data points.

Flashcard 51: At $60$ miles per hour, how far is traveled in $2.5$ hours?

Answer: $150$ miles. $60 \times 2.5 = 150$ miles

Flashcard 52: Worker A completes $\frac{1}{4}$ of a job per hour; how many hours for the whole job?

Answer: $4$ hours. Rate is $\frac{1}{4}$ job per hour, so 4 hours total.

Flashcard 53: What is the key step in solving mixture word problems?

Answer: Set up an equation based on total quantity or concentration. Balance total amounts or percentages to find unknowns.

Flashcard 54: At $60$ miles per hour, how far is traveled in $2.5$ hours?

Answer: $150$ miles. $60 \times 2.5 = 150$ miles

Flashcard 55: How do you express 'three more than a number' algebraically?

Answer: $x + 3$. Addition of 3 to the unknown variable $x$.

Flashcard 56: Identify the translation: "three times the sum of $x$ and $4$" becomes what expression?

Answer: $3(x+4)$. First add $x$ and 4, then multiply by 3.

Flashcard 57: Which operation is essential in work-related word problems?

Answer: Addition of rates. Combined work rates determine completion time.

Flashcard 58: What equation represents 'a number decreased by 8 is 12'?

Answer: x - 8 = 12. Subtraction translates to the minus operation algebraically.

Flashcard 59: Identify the equation to solve: 'A number is 5 more than twice another number.'

Answer: Equation: $x = 2y + 5$. Translates 'twice another' as $2y$ and 'is 5 more than' as $+5$.

Flashcard 60: Find the original price if an item costs $36$ after a $10\%$ discount.

Answer: $40$. $36 = 0.90x$, so $x = 40$

Flashcard 61: How is 'ten more than twice a number' expressed algebraically?

Answer: 2x + 10. Multiply by 2 first, then add 10.

Flashcard 62: What is the formula for simple interest on principal $P$ at rate $r$ for time $t$ (years)?

Answer: $I=Prt$. Interest equals principal times rate times time.

Flashcard 63: What is the first step in solving a word problem involving percentages?

Answer: Identify the given and unknown quantities. Essential first step to organize the problem structure.

Flashcard 64: Which mathematical operation is used when distributing items evenly?

Answer: Division. Splits quantities into equal parts among recipients.

Flashcard 65: A fair coin is flipped twice; what is the probability of exactly one head?

Answer: $\frac{1}{2}$. Two outcomes (HT, TH) out of four possible.

Flashcard 66: What is the first step in solving mixture problems?

Answer: Identify the components and their quantities. Organize what's being mixed and in what amounts.

Flashcard 67: What is the mixture formula for total amount when combining $a$ units and $b$ units?

Answer: $\text{total amount}=a+b$. Total quantity is sum of individual amounts.

Flashcard 68: What equation represents 'Four times a number is reduced by 8 to get 16'?

Answer: Equation: $4x - 8 = 16$. Translates 'reduced by 8' as subtraction from $4x$.

Flashcard 69: Translate 'the difference between a number and 5 is 10' into an equation.

Answer: x - 5 = 10. Subtraction relationship between two quantities equals 10.

Flashcard 70: How do you solve a percent increase word problem?

Answer: $\text{New Value} = \text{Original Value} \times (1 + \text{Percent Increase})$ . Multiplies original by $(1 + \text{decimal percent})$ for increase.

Flashcard 71: What equation represents 'the sum of a number and 11 is 25'?

Answer: x + 11 = 25. Addition equation with variable and constants.

Flashcard 72: How is 'ten more than twice a number' expressed algebraically?

Answer: 2x + 10. Multiply by 2 first, then add 10.

Flashcard 73: A store charges $5 fixed plus $2 per item; what is the cost for $7$ items?

Answer: $19. $5 + 7(2) = 19$ dollars total.

Flashcard 74: Identify the equation for 'Three times a number equals 18.'

Answer: Equation: $3x = 18$. Direct translation of 'three times' as multiplication.

Flashcard 75: What is the approach to solve a rate of work problem with two workers?

Answer: Add their rates to find total rate. Combined rates equal sum of individual work rates.

Flashcard 76: What is the definition of average rate of change from $x=a$ to $x=b$ for $f(x)$?

Answer: $\frac{f(b)-f(a)}{b-a}$. Change in function value divided by change in input.

Flashcard 77: Identify the translation: "three times the sum of $x$ and $4$" becomes what expression?

Answer: $3(x+4)$. First add $x$ and 4, then multiply by 3.

Flashcard 78: What is the key step in solving mixture word problems?

Answer: Set up an equation based on total quantity or concentration. Balance total amounts or percentages to find unknowns.

Flashcard 79: How do you express 'three-fifths of a number' in algebraic terms?

Answer: $\frac{3}{5}x$. Fraction multiplied by variable shows partial amount.

Flashcard 80: What equation represents 'Seven less than a number is 12'?

Answer: Equation: $x - 7 = 12$. Translates 'seven less than' as $x - 7$.

Flashcard 81: A die is rolled once; what is the probability of rolling a multiple of $3$?

Answer: $\frac{1}{3}$. Multiples of 3 are 3 and 6, so 2 out of 6.

Flashcard 82: Identify the translation: "$5$ more than a number $x$" becomes what expression?

Answer: $x+5$. Add 5 to the number $x$.

Flashcard 83: Which operation is used to find the total in consecutive number problems?

Answer: Addition. Sums consecutive integers to find their total.

Flashcard 84: What equation represents 'Four times a number is reduced by 8 to get 16'?

Answer: Equation: $4x - 8 = 16$. Translates 'reduced by 8' as subtraction from $4x$.

Flashcard 85: How do you express 'the sum of twice a number and 8' algebraically?

Answer: 2x + 8. Addition of doubled variable and constant term.

Flashcard 86: Identify the equation for 'The quotient of a number and 5 is 3.'

Answer: Equation: $\frac{x}{5} = 3$. Translates 'quotient' as division operation.

Flashcard 87: What is the first step in solving a time, rate, and distance problem?

Answer: Identify given values for time, rate, or distance. Organize known information before applying $d = rt$ formula.

Flashcard 88: What equation represents 'Three times a number decreased by 4 equals 11'?

Answer: Equation: $3x - 4 = 11$. Translates 'three times' as $3x$ and 'decreased by 4' as $-4$.

Flashcard 89: A $5$ lb nut mix costs $4/lb and $6/lb nuts are combined; if total cost is $26, how many pounds at $6/lb?

Answer: $3$ lb. Let $x$ be pounds at $6: $2(4) + x(6) = 26$, so $x = 3$.

Flashcard 90: Identify the translation: "$7$ less than a number $x$" becomes what expression?

Answer: $x-7$. Subtract 7 from the number $x$.

Flashcard 91: How do you write 'the ratio of a number to 7' as an expression?

Answer: $\frac{x}{7}$. Division creates the fractional relationship to 7.

Flashcard 92: How do you translate 'twice a number' into an algebraic expression?

Answer: 2x. Multiply by 2 to represent doubling the value.

Flashcard 93: What does the phrase "per" usually indicate in an ACT word problem expression?

Answer: "Per" usually means division, as in $60$ miles per $2$ hours is $\frac{60}{2}$. This indicates division to find rate in word problems.

Flashcard 94: How do you calculate the selling price with a discount?

Answer: Selling Price = Original Price × (1 - Discount Rate). Multiplies by $(1 - \text{decimal discount})$ to reduce price.

Flashcard 95: A rectangle has area $48$ and width $6$; what is its length?

Answer: $8$. Area is length times width, so $l = \frac{48}{6} = 8$.

Flashcard 96: What is the key step in solving rate problems involving two objects?

Answer: Set equations based on relative rates. Compare speeds to determine relative motion.

Flashcard 97: How do you express 'three more than a number' algebraically?

Answer: x + 3. Addition of 3 to the unknown variable $x$.

Flashcard 98: A price decreases from $80$ to $68$ dollars; what is the percent decrease?

Answer: $15\%$. $\frac{80-68}{80} \times 100\% = 15\%$

Flashcard 99: Identify the translation: "the difference of $x$ and $4$" becomes what expression?

Answer: $x-4$. Subtract the second number from the first.

Flashcard 100: What is the work-rate equation for two workers with rates $r_1$ and $r_2$ working together?

Answer: $r_1+r_2=r_{\text{total}}$. Combined work rates add when working together.