This guide focuses on explaining how to do the hardest ACT math problems. There is currently nothing like it available.
This comprehensive guide is designed to help students master the hardest ACT math problems using proven strategies and step-by-step explanations. Whether you’re aiming for a perfect score or looking to improve your performance, this resource focuses on advanced techniques, real ACT math practice questions, and expert insights.
Unlike other resources, this guide is built specifically for high-achieving students who want to tackle hard ACT math questions with confidence. While the guide is still under development, it already provides valuable content covering the most challenging topics in the ACT math test, helping students strengthen their problem-solving skills and boost their scores.
This study guide covers almost everything likely to appear on the exam. If you study this guide thoroughly, you should be well prepared. This guide is more accessible to a wider range of students than the categorized problems in this workbook, which are generally extremely challenging. It is recommended that students with average proficiency start with the common problems, and then move on to this study guide. Each problem is marked with its level of difficulty, along with notes on especially common questions and types of trap problems.
3 What is the factorization of $x^2-3x-28$? $x^2+4x-7x- 28\implies x(x+4)-7(x+4)=(x+4)(x-7)$. You could also use the quadratic formula and reverse engineer the factorization.
4 If $x-3$ is a factor or $3x^2-4x+k$, what is $k$? Substitute in 3, set equal to 0 and solve. $3\cdot 3^2-4\cdot 3+k=0\implies 27-12+k=0\implies k=-15$.
4 What is the factorization of $3x^3+2x^2-5x$? $x(3x^2+2x-5)\implies x(3x^2-3x+5x-5)\implies x(3x(x-1)+5(x-1))\implies x(x-1)(3x+5)$. It might be possible to use the quadratic formula or plug in answer choices.
The basic mean problem involves adding all the values and dividing by the number of values.
2 What is the mean of 17, 11, 23, 8, 18, 21, 24, and 14?$\dfrac{17+11+23+8+18+21+24+14}{8}=$$\dfrac{136}{8}=17$.
The median is the middle value when the values are in sorted order. It is falling into a trap to take the middle without first sorting the values. If there are an even number of values, the median is the average of the two middle values. If the data is grouped in intervals, you need to add the number of values in the intervals from either direction and determine in which interval the median must be.
3 The median of 11 numbers is 17 and no two numbers are the same. How many of the numbers are greater than 17? $\dfrac{10}{2}=5$.
4-C What is the median of 17, 11, 23, 8, 18, 21, 24, and 14? First sort the numbers: 8, 11, 14, 17, 18, 21, 23, and 24. Since there is an even number of values, the median is the average of the two middle values, 17 and 18: $\dfrac{17+18}{2}=17.5$.
To find the weighted average, multiply each value by the number of elements (or frequency) associated with that value, then add all those products, and divide that by the sum of all the frequencies.
4A restaurant had the following ratings 37 5s, 17 4s, 8 3s, 5 2s, and 11 1s. What was the weighted average?$\dfrac{37\cdot 5+17\cdot 4+8\cdot 3+5\cdot 2+11\cdot 1}{37+17+8+5+11}=$$\dfrac{185+68+24+10+11}{78}=\dfrac{298}{78}\approx 3.82$.
For "need on test" problems you take the sum of your current test scores, usually by taking the average of your current test scores times the number of current tests, and add that to the unknown $x$, and then divide by the number of tests including the one with the unknown score; set that equal to the desired score, and solve for $x$. There could be fairly difficult problems in which the current average or desired score is unknown, and you need to get an expression in terms of variables.
4 Bob has a 78, 83, 86, 90, and 92 on 5 equally weighted tests. What is the minimum score he needs on the $6^{th}$ test to raise his average by 3 points? Current average $=\dfrac{78+83+86+90+92}{5}=\dfrac{429}{5}=85.8$. Therefore, he needs an 88.8. $\dfrac{429+x}{6}=88.8\implies 429+x=532.8\implies x=103.8$ or 104.
4 The average of 5 numbers is 76. A new list of numbers has all the same numbers except the $5^{th}$ number is changed from 67 to 82. What is the average of the new list? $\dfrac{82-67}{5}=3$, $76+3=79$.
4, CA student has a 78, 74, and 82 on 3 tests. What is the minimum he needs on the $4^{th}$ equally weighted test to average 80 for all 4 tests? $\dfrac{78+74+82+x}{4}=80\implies \dfrac{234+x}{4}=80\implies 234+x=320\implies x=86$.
4A student has an average of 78 on 6 equally weighted tests. What does he need on the $7^{th}$ test to average 80 for all 7? $\dfrac{78\cdot 6+x}{7}=80\implies 468+x=560\implies x=92$.
5A student has an average of $x$ on 8 equally weighted tests. If the lowest grade is removed, his average is $y$. What was the lowest grade? The total is $8x$ on the 8 tests. $\dfrac{8x-w}{7}=y$. Now solve for $w$, the lowest score. $8x-w=7y\implies w=8x-7y$.
4 , CWhat is the difference between the mean and the median of the squares of the integers from 1 to 4? The squares of those integers are 1, 4, 9, and 16. Since there are an even number of elements and the data is sorted, the median is the average of the two middle elements: $\dfrac{4+9}{2}=6.5$. The mean is the average of all the elements: $\dfrac{1+4+9+16}{4}=7.5$. Therefore, the difference is 1.0.
4 What is the product of the mean and median of the first 7 prime numbers? 2, 3, 5, 7, 11, 13, 17. Median is the middle value in sorted order, 7. Mean is $$\dfrac{2+3+5+7+11+13+17}{7}=\dfrac{58}{7}$, $7\cdot \dfrac{58}{7}=58$$
Divide the percent by 100 to convert it to a decimal, and then multiply that decimal by the number you are taking a percentage of.
2 What is 150% of 236? $236\cdot 1.5=354$.
2 What is a 15% tip on a \$65 restaurant bill? $65\cdot 0.15=\$9.75$.
3 ,C What is 6% of $2.52\times 10^5$? Multiplying, $0.06\cdot 2.52\cdot 10^5=0.1512\cdot 10^5=1.512\cdot 10^4$. You can also enter the whole thing with your calculator. A key step is converting 6% to 0.06.
To find percent of percent of a number, multiply that number by each percentage converted to a decimal.
3 What is 30% of 40% of 90? $0.3\cdot 0.4\cdot 90=10.8$.
Take the difference divided by the original value and then multiply by 100.
3 The price of a motel room was decreased for \$80 per night to per \$70 night . What was the percent decrease?$80-70=10$. $\dfrac{10}{80}\cdot 100=12.5\%$.
3 A motel room costs \$80 per night, but its price was increased by $35\%$ on a high demand weekend. What was the higher price? $80\cdot 1.35=\$108$.
3 A shirt was originally priced at \$30, but is sold at $30\%$ off, but with a $7\%$ sales tax on the sale price. How much did the customer pay including tax?$30\cdot 0.7=21$ (0.7 because $1-0.3$ for 30%). $21\cdot 1.07=\$22.47$.
For compound percent (one percent change followed by another), add or subtract each percentage change as a decimal from 1, depending on whether it represents an increase or decrease. Then multiply the resulting decimal. Finally, subtract 1 from the product, and multiply by 100 to convert the result back to a percentage.
5 If the price of a stock goes up 10% in 2022 and 30% in 2023, what is the total percent increase in the stock price?$1.1\cdot 1.3=1.43$. $(1.43-1)\cdot 100=43\%$.
5 If the length, width, and height of a box are each increased by 20%, what is the percent increase in its volume?${1.2}^3=1.728$, so approximately 73%.
5 The length of a rectangle is increased by 40% and its width is decreased by 20%. By what percent is the area of the new rectangle larger than the original one?$1.4\cdot 0.8=1.12$. $(1.12-1)\cdot 100=12\%$.
4 The price of an item was reduced by 20% then by 30%. The fully discounted price is what percent of the original price?$0.8\cdot 0.7=0.56$ or $56\%$.
It is helpful to know the midpoint formula $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$. You can also understand the concept, which is the average of the $x$-coordinates and the average of the $y$-coordinates. These problems should be solvable by understanding the concepts without knowing the formula.
1 What is the midpoint on the number line between 7 and 23? Just average the numbers: $\dfrac{7+23}{2}=15$.
2,F,C,C What is the midpoint between point $A(2,-5)$ and $B(10,7)$? Applying the midpoint formula, $\left(\dfrac{2+10}{2}, \dfrac{-5+7}{2}\right)=(6,1)$.
3, F,C,C $M$ is the midpoint of $AB$. $A$ is $(2,1)$ and $M$ is $(5,9)$. What is $B$? There are various ways to reason this, but the algebraic approach is to substitute in the answer $M$ and the endpoint $A$ and then solve for $B$. $5=\dfrac{2+x}{2}\implies 10=2+x\implies x=8$. $9=\dfrac{1+y}{2}\implies 18=1+y\implies y=17$. Therefore, the answer is $(8,17)$.
For distance problems, the main thing is to know the distance formula. If you do not know the formula, then you can also use the Pythagorean theorem directly, and the formula can be derived from that theorem. The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
2 , F , C What is the distance between $(-2, 5)$ and $( -7,-6)$? Applying the distance formula,$d=\sqrt{(-2-(-7))^2+(5-(-6))^2}=$ $\sqrt{5^2+11^2}=\sqrt{25+121}=\sqrt{146}$.
2 , F What is the distance between $(0,30)$ and $(30, 0)$?$\sqrt{(30-0)^2+(0-30)^2}=\sqrt{900+900}=\sqrt{1800} \approx 42$. You could also use that the ratios of a $45-45-90$ triangle are $1-1-\sqrt{2}$ or use trigonometry with your calculator.
2 , C Carlos gets \$18/hour plus time and a half for overtime. If he worked 54 hours last week, how much did he get paid?$18\cdot 40+27\cdot 14=720+378=\$1098$.
3 Abdul earns \$28 per hour for the first 40 hours a week and time and a half for overtime. If he earned \$1540 for the week, how many hours did he work? Full-time pay is $28\cdot 40=1120$. Overtime pay is $1540-1120=420$. $\dfrac{420}{28\cdot 1.5}=10$. $40+10=50$ hours.
3 A rental car company charges \$60/day plus 45 cents per mile. What would the charge for 5 days and 400 miles be?$60\cdot 5+0.45\cdot 400=300+180=\$480$.
3 , C Abdul and Julio were painting a house. They started with 6 gallons of paint. The first day Abdul used $1\dfrac{5}{8}$ gallons and Julio used $2\dfrac{1}{4}$ gallons. How many gallons were left?$$6-1\dfrac{5}{8}-2\dfrac{1}{4}=6-\dfrac{13}{8}-\dfrac{9}{4}=\dfrac{48-13- 18}{8}=\dfrac{17}{8}=2\dfrac{1}{8}.$$
3 If $x+2\dfrac{1}{4}=4\dfrac{5}{8}$? $x=4\dfrac{5}{8}-2\dfrac{1}{4}=2\dfrac{3}{8}$.
4 For what integers does $\dfrac{5}{x}$ lie between $\dfrac{1}{4}$ and $\dfrac{1}{3}$? 16, 17, 18, 19. By your calculator or other methods.
4 Which of the following is equivalent to$$\dfrac{\dfrac{a}{b}+1}{\dfrac{1}{a}+\dfrac{1}{b}}\text{ ? } \dfrac{\dfrac{a+b}{b}}{\dfrac{b+a}{ab}}\implies \dfrac{(a+b)ab}{(a+b)b}=a.$$
4 Which of the following is equivalent to $\dfrac{\dfrac{x}{3} + \dfrac{2}{5}}{\dfrac{1}{4}+\dfrac{1}{3}}$? $$\dfrac{\dfrac{5x+6}{15}}{\dfrac{7}{12}}\implies \dfrac{(5x+6)12}{15\cdot 7}=\dfrac{4(5x+6)}{35}.$$
3 Which of the following is equivalent to $\dfrac{2}{x}+\dfrac{3}{5}$? The least common denominator is $5x$.$\dfrac{2}{x} \cdot\dfrac{5}{5}+\dfrac{3}{5} \cdot \dfrac{x}{x}=\dfrac{10+3x}{5x}$.
3 Which of the following is equivalent to$\dfrac{1}{x-2}+\dfrac{1}{x+2}$? $\dfrac{x+2+x-2}{(x-2)(x+2)}\implies \dfrac{2x}{x^2-4}$.
3 Which of the following is equivalent to$\dfrac{1}{x}+\dfrac{5}{x^3}$? $\dfrac{x^2+5}{x^3}$.
Generally, factor the numerator and denominator. Then cancel like terms.
4 Which is equivalent to$\dfrac{x^2-2x-35}{x^2-25}$? $\dfrac{(x-7)(x+5)}{(x+5)(x-5)}=\dfrac{x-7}{x-5}$.This also can be solved by substituting a number for $x$ in the problem and in the answer choices.
3 , C What is the least common denominator of $\dfrac{3}{8}$, $\dfrac{5}{12}$, and $\dfrac{11}{28}$? $8=2\cdot 2\cdot 2$, $12=2\cdot 2\cdot 3$, $28=2\cdot 2\cdot 7$. The least common denominator is $2\cdot 2\cdot 2\cdot 3\cdot 7=168$
3 , C One sign flashes every 6 seconds and another every 8 seconds. At a certain instant, they flash at the same time. How many seconds until they flash at the same time again? This is asking for the least common multiple. Take $6=2\cdot 3$ and $8=2\cdot 2\cdot 2$. Take all factors in either: $2\cdot 2\cdot 2\cdot 3=24$.
3 , C , C What is the least common multiple of 50, 60, and 90? $50=2\cdot 5\cdot 5$, $60=2\cdot 2\cdot 3\cdot 5$, $90=3\cdot 3\cdot 2\cdot 5$. Therefore, the least common multiple is $2\cdot 2\cdot 3\cdot 3\cdot 5\cdot 5=900$.
2 , C What is $(4x+7)(4x-7)$?$16x^2-28x+28x-49=16x^2-49$.
2 , C What is the area of a rectangle with length $2x+3$ and width $x+5$? $(2x+3)(x+5)=2x^2+10x+3x+15=2x^2+13x+15$.
3 Which is equivalent to ${\left(\dfrac{x}{2}-4\right)}^2$?$\dfrac{x^2}{4}-2x-2x+16=\dfrac{x^2}{4}-4x+16$.
3 Which is equivalent to $\left(a+\sqrt{b}\right)\left(a+3\sqrt{b}\right)$? $a^2+3a\sqrt{b}+a\sqrt{b}+3b=a^2+4a\sqrt{b}+3b$.
$a^2+b^2=c^2$, where $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse.
2 A rectangle is 24 feet long and 10 feet wide. What is the length of its diagonal? $10^2+24^2=d^2\rightarrow 100+576=d^2\rightarrow 676=d^2\rightarrow d=26$ feet.
3 If one leg of a right triangle is $\sqrt{7}$ and the hypotenuse is 4, what is the length of the other leg? ${\left(\sqrt{7}\right)}^2+b^2=4^2\rightarrow 7+b^2=16\rightarrow b^2=9\rightarrow b=3$.
1 What is $7-3(x-5)$? $7-3x+15=22-3x$.
1 What is $(3a-3b+7c)-(2a+4b-6c)$? $3a-3b+7c-2a-4b+6c=a-7b+13c$.
2 , C Which of the following expressions is equivalent to $5(x+4)-2(2x-3)$? $5x+20-4x+6=x+26$
4 Which of the following expressions is equivalent to $\dfrac{x^2+8x+15}{x+3}+2x+4$? $\dfrac{(x+5)(x+3)}{x+3}+2x+4=x+5+2x+4=3x+9$.
1 , C What is the smallest positive integer greater than $\sqrt{130}$? 12. With your calculator $\sqrt{130}\approx 11.4$, so 12. Rounding down to 11 would be falling into a trap. It is possible to do this problem if you know the perfect squares without using your calculator.
This is a fairly common type of relatively easy problems on the exam.$x^a\cdot x^b=x^{a+b}$, $(x^a)^b=x^{ab}$, $(2x^4)^3=8x^{12}$.
2 , C Which of the following expressions is equivalent to $(3x^5y^3)(8x^3y^2)$? Grouping like terms, $(3\cdot 8)(x^5\cdot x^3) (y^3\cdot y^2) =24x^8y^5$. When multiplying variables raised to exponents, add the exponents.
2 , C Which of the following is equal to ${\left(x^4\right)}^{24}$? $x^{96}$ When taking the power of the power, you multiply the exponents.
2 , C Which of the following is equivalent to $(2x^2)^4$? $2^4\cdot (x^2)^4=16x^8$. When taking the power of the power, multiply exponents.
2 , C , C Which of the following is equivalent to $(5x^3)\cdot (4x^8)$? $20x^{11}$.
3 , C The expression $\dfrac{4a^3b^5c^2}{10a^4b^3c^5}$ is equivalent to what? $\dfrac{4}{10}\cdot \dfrac{a^3}{a^4}\cdot \dfrac{b^5}{b^3}\cdot \dfrac{c^2}{c^5}=\dfrac{2b^2}{5ac^3}$.
3 $\dfrac{\dfrac{x^5}{x^8}}{\dfrac{x^7}{x^2}}=$? Flip the fraction you are dividing by: $\dfrac{x^5\cdot x^2}{x^8\cdot x^7}=\dfrac{x^7}{x^{15}}=\dfrac{1}{x^8}$.
3 $(2x^4)^3(3x^2)^2=$? $8x^{12}\cdot 9x^4=72x^{16}$.
3 $2^4x^5y^{-3}3^{-2}=$? $\dfrac{16x^5}{9y^3}$.
3 How many seconds would it take to travel 30 miles at 50 miles per hour? $\dfrac{30}{50}=\dfrac{3}{5}~\text{hour}\implies \dfrac{3}{5}\cdot 60\cdot 60=2160$ seconds.
3 540 square feet is how many square yards? Each square yard is $3\times 3=9$ square feet. Therefore, $\dfrac{540}{9}=60$ square yards.
3 A 120 feet by 210 feet field is how many square yards? That is 40 yards by 70 yards, so 2800 square yards. You could also take $\dfrac{120\cdot 210}{9}=2800$.
3 What is the minimum number of 6-inch by 9-inch tiles needed to cover a 12 feet by 15 feet floor? The area of the floor is $12\times 15=180$ square feet. Each tile is $\dfrac{1}{2}\cdot \dfrac{3}{4}=\dfrac{3}{8}$ square feet. Therefore, $\dfrac{180}{\dfrac{3}{8}}=480$ tiles. You could also calculate that each tile is 54 square inches and find the area of the floor in square inches, but that method is more time consuming.
3 , C A board that is 11 feet 2 inches long is cut into two equal parts. How long is each part in feet and inches? Half of 11 feet is 5 feet 6 inches. Half of 2 inches is 1 inch, so each part is 5 feet 7 inches. Or convert to inches: $11\cdot 12+2=134$. Half of 134 is 67. $\dfrac{67}{12}=5$ with remainder 7, so 5 feet 7 inches.
3 What is 38 million in scientific notation? One million is ${10}^6$. 38 is $3.8\times 10$. Therefore, $3.8\times 10\cdot {10}^6=3.8\times {10}^7$.
3 , C What is 0.00000456 in scientific notation? The 4 is the $6^{th}$ digit after the decimal point, so $4.56\times {10}^{-6}$.
4 $5.5\times 10^{3x+2}\cdot 4\cdot 10^{-5}=220$. What is $x$? $22\cdot 10^{3x+2}\cdot 10^{-5}=220\implies 2.2\times 10\cdot 10^{3x-3}=2.2\cdot 10^2\implies 2.2\times 10^{3x-2}=2.2\cdot 10^2\implies 3x-2=2\implies 3x=4\implies x=\dfrac{4}{3}$.
3 There are $3\times 10^{16}$ molecules in an $8\times 10^{10}$ cubic cm box, how many molecules are there per cubic cm? $\dfrac{3\times 10^{16}}{8\times 10^{10}}=\dfrac{3}{8}\times {10}^{6}= 0.375 \times 10^{6} = 3.75 \times 10^{3} $.
3 What is sum of $45,000+78,000$ in scientific notation? $123,000=1.23\times {10}^5$.
3 What is the sum of $7.3\times 10^3$ and $2.1\times 10^4$? $7.3\times 10^3=0.73\times 10^4$. $2.1\times 10^4+0.73\times 10^4=2.83\times 10^4$.
3 $3.74\times {10}^{-23}$ would have how many zeros after the decimal if expressed as $0.0\ldots374$? The 3 starts at the $22^{nd}$ place, so 21 zeros.
1 , C If $7x-11=4x+7$, then $x=$?$3x=18\implies x=6$.
2 , C If $\dfrac{4x}{5}-3=17$, what does $x$ equal? $\dfrac{4x}{5}=20\implies x=20\cdot \dfrac{5}{4}=25$.
2 , C Which of the following is inequalities is equivalent to $5x-3>2x+4$? $3x>7$, $x>\dfrac{7}{3}$.
2 $\dfrac{5x}{3}+8=20\implies \dfrac{5x}{3}=12\implies$$x=12\cdot \dfrac{3}{5}\implies x=\dfrac{36}{5}$.
3 For what value of $x$ is $\dfrac{5+x}{2+x}=\dfrac{3}{5}$? $(5+x)5=(2+x)3\cdot 25+5x=6+3x\implies 2x=-19\implies x=\dfrac{-19}{2}$.
3 For what value of $x$ does $4.2x+2.4=1.7x+5.9$? $2.5x=3.5\implies x=1.4$.
3 $\dfrac{2x}{3}+\dfrac{3}{4}=\dfrac{5}{2}$. What is $x$? $\dfrac{2x}{3}=\dfrac{7}{4}\implies x=\dfrac{7}{4}\cdot \dfrac{3}{2}\implies x=\dfrac{21}{8}$.
3 A flight was scheduled to depart at 9:37~AM but was 476 minutes late. When did it depart? 7 hours 56 minutes or 8 hours minus 4 minutes 5:33 PM.
3 17 ft 3 inches is how much longer than 12 ft 8 inches? 5 feet minus 5 inches or 4 feet 7 inches.
4 , CA train left Atlanta at 7:46 PM and arrived in Baltimore at 3:12 AM the next day. How many hours and minutes did the trip take? 7 hours to 2:46 AM and another 26 minutes to 3:12 AM, so 7 hours and 26 minutes.
The sum of 2 sides of a triangle is always greater than the third side.
3 If two sides of a triangle have lengths 7 and 10, the third side must be in what range? $10-7 < x < 10+7\implies 3< x< 17$.In a triangle the larger side is opposite the larger angle.
There are many different types of number theory problems on the exam. They tend to be fairly difficult and intended to test reasoning ability.
The units digits of powers of numbers ending in 2, 3, 7 and 8 repeat every 4. Therefore, determine the pattern and take the exponent modula 4.
4 If the units digit of $73^{235}$ is 7, what is the units digit of $73^{238}$? $7\cdot 3^3=7\cdot 27=189$, so 9.
5 What is the units digit of $78^{138}$? $8^1=8$, $8^2=4$, $8^3=2$, $8^4=6$, all mod 10. Now 138 mod $4=2$, so we use $8^2$, which is 4.
4 If $x$ is a positive integer, the sum of $8x$ and $9x$ is always divisible by what number? $8x+9x=17x$, so 17.
4 What is the least positive number which has remainder 5 when divided by 7 and remainder 7 when divided by 11? Numbers with remainder 5 when divided by 7: 5, 12, 19, 26, 33, 40; numbers with remainder 7 when divided by 11: 7, 18, 29, 40. Therefore, the answer is 40.
5 What is the $573^{rd}$ digit to the right of the decimal in $.\overline{3756}$? 573 mod $4=1$, so take the first digit in the sequence, 3. You can accomplish modular arithmetic by dividing like in elementary school and taking the remainder or dividing by 4 with your calculator and then multiplying the decimal part by 4.
5 How many numbers between 1 and 200 are divisible by 2, 3, and 7? They must be divisible by $2\cdot 3\cdot 7=42$. $42$, $84$, 126, 168. Therefore, 4 numbers.
5 What percent of even numbers between 2 and 40, inclusive, have units digits twice the tens digit? 12 24 36, so 3 out of 20, so 15%.
5 What is the largest $3$-digit number divisible by $7$ and $11$. It needs to be divisible by $77$. $77\cdot 12=924$.
5 How many prime numbers are there between 50 and 80? Taking odd numbers 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79. Remove numbers divisible by 3 (sum of digits divisible by 3) and numbers divisible by 5 (if the last digit is 5). 53 59 61 67 71 73 77 79. Remove numbers divisible by 7: 53 59 61 67 71 73 79, so 7 prime numbers.
2 , C , F In the standard $xy$-coordinate plane, what is the slope of a line containing $(5,7)$ with $y$-intercept of $-2$? The points are $(5,7)$ and $(0,-2)$. The slope formula is $\dfrac{y_2-y_1}{x_2-x_1}$. Therefore, $\dfrac{7-(-2)}{5-0}=\dfrac{9}{5}$ is the slope.
2 , C , F In the standard $xy$-coordinate plane, what is the slope of the line $5x-11y=17$? Convert to slope-intercept form: $11y=5x-17\implies y=\dfrac{5x}{11}- \dfrac{17}{11}$, so the slope is the $x$-coefficient, $\dfrac{5}{11}$.
2 , C , C, F What is the slope of a line through $(-5,7)$ and $(4,1)$ in the standard $xy$-coordinate plane? Using the slope formula $\dfrac{y_2-y_1}{x_2- x_1}$. $\dfrac{7-1}{-5-4}=\dfrac{6}{-9}=\dfrac{-2}{3}$.
2, C What is the slope of$y-17=\dfrac{3}{4}(2x+5)$?Distributing, $y-17=\dfrac{3x}{2}+\dfrac{15}{4}$.The slope is the coefficient of $x$, so it is $\dfrac{3}{2}$.
To find the $x$-intercept, set $y$ equal to 0 and solve for $x$. To find the $y$-intercept, set $x$ equal to 0 and solve for $y$. If an equation is in slope-intercept form, that is solved for $y$, the $y$-intercept is the constant term.
2 What is the $x$-intercept of $3x+5y=11$?$3x+5\cdot 0=11\implies 3x=11\implies x=\dfrac{11}{3}$.
4 What is the $x$-intercept of the line through $(2,5)$ and $(4,8)$?\\ Slope $=\dfrac{8-5}{4-2}=\dfrac{3}{2}$. $5=2\cdot \dfrac{3}{2}+b\implies b=2$. Therefore, $y=\dfrac{3x}{2}+2$. $0=\dfrac{3x}{2}+2\implies -2=\dfrac{3x}{2}\implies x=\dfrac{-4}{3}$.
4 What is the $x$-intercept of the line through $(3,8)$ with slope of 2? $y=mx+b\implies 8=2\cdot 3+b\implies b=2$. Therefore, $y=2x+2$. Setting $y=0$, $0=2x+2\implies x=-1$.
3 What is the equation of the line through $(3,5)$ with a slope of $\dfrac{1}{2}$? $y=mx+b$. $5=3\cdot \dfrac{1}{2}+b\implies 5=\dfrac{3}{2}+b\implies b=\dfrac{7}{2}$. $y=\dfrac{x}{2}+\dfrac{7}{2}$.
4 , F What is the equation of the line through $(3,1)$ and $(6,5)$? Slope $=\dfrac{5-1}{6-3}=\dfrac{4}{3} y=mx+b$. $1=3\cdot \dfrac{4}{3}+b\implies\\ 1=4+b\implies b=-3$. $y=\dfrac{4x}{3}-3$.
4 Bob drove 500 miles in 10 hours. By averaging 10 mph faster, how many minutes would he save? $\dfrac{500}{10}=50$ mph. 10 mph faster is 60 mph. $\dfrac{500}{60}=\dfrac{25}{3}$. $10-\dfrac{25}{3}=\dfrac{5}{3}$ hours. $\dfrac{5}{3}\cdot 60=100$ minutes.
4 Steve drove at 40 mph for 10 minutes and 60 mph for 15 minutes. What was his average speed for the 25 minutes? Distance traveled = $40\cdot \dfrac{1}{6}+60\cdot \dfrac{1}{4}=\dfrac{20}{3}+15=\dfrac{65}{3}$ miles. Time spent traveling = 25 minutes $=\dfrac{25}{60}$ hours $=\dfrac{5}{12}$ hours. Therefore, the rate or speed is $\dfrac{D}{T} = \dfrac{\dfrac{65}{3}}{\dfrac{5}{12}}=\dfrac{780}{15}=52$ mph.
4 Sally travels for 5 hours at 50 mph to get there. Due to heavy traffic, for the first 2 hours of her return trip, she averages 20 mph. What must her average speed for the rest of her return trip be for her to finish the return trip in 6 hours total? $5\cdot 50=250$, so the one-way trip is 250 miles. She has traveled $20\cdot 2=40$ miles so far. Therefore, she has $250-40=210$ miles to go. She has $6-2=4$ hours left. Therefore, the average speed must be $\dfrac{210}{4}=52.5$ mph.
3 A 5.3 kilometer cab ride took 12 minutes. What was the average speed in mph (1 mile $=1.6$ kilometers)? 12 minutes $=\dfrac {12}{60}$ hours $= \dfrac {1}{5}$ hours. $\dfrac{5.3}{\dfrac{1}{5}}=26.5$ kilometers/hour. $\dfrac{26.5}{1.6}\approx 16.6$~mph.
4 An object traveled 300 feet in 2.5 seconds. What was its average speed in mph (1 mile $=5,280$ feet)? $\dfrac{300}{2.5}=120$ feet/second. $120\cdot 60\cdot \dfrac{60}{5280}\approx 81.8$ ~mph.
4 Darrel ran 20 miles in $2\dfrac{1}{2}$ hours. What is the average number of minutes it took him to run one mile? $\dfrac{20}{\dfrac{5}{2}}= 8$ mph. Since one hour is 60 minutes, the time in minutes is $\dfrac{60~\text{minutes}}{8~\text{mph}}=7.5$ minutes.
4 Vehicle $X$ gets 20 mpg and Vehicle $Y$ gets 32 mpg. How many more gallons will vehicle $X$ consume than vehicle $Y$. both traveling 2000 miles? $\dfrac{2000}{20}=100$ gallons. $\dfrac{2000}{32}=62.5$ gallons. $100-62.5=37.5$ gallons.
4 Robert ran a 6-mile cross country course in 35 minutes; Willie ran the same course in 40 minutes. What is the difference in their speeds in mph? $\dfrac{6}{\dfrac{40}{60}}=9$ mph. $\dfrac{6}{\dfrac{35}{60}}\approx 10.28$ mph. Therefore, the difference is $1.28$ mph.
4 The marbles in a bowl are $\dfrac{1}{3}$ red, $\dfrac{1}{4}$ blue, $\dfrac{1}{5}$ yellow, and the other $26$ green. How many marbles in the bowl? The portion that are green are $1-\dfrac{1}{3}-\dfrac{1}{4}-\dfrac{1}{5}=\dfrac{60}{60}- \dfrac{20}{60}-\dfrac{15}{60}- \dfrac{12}{60}=\dfrac{13}{60}=\dfrac{26}{120}$, so there are $120$ marbles in the bowl.
4 Bob did $\dfrac{1}{3}$ of a job and Steve did $\dfrac{1}{4}$ of the job. Then Robert completed the job in 10 hours. If they all worked at the same rate, how many hours did the job take? The portion remaining after Bob and Steve's initial work is $1-\dfrac{1}{3}-\dfrac{1}{4}=\dfrac{12}{12}-\dfrac{4}{12}- \dfrac{3}{12}=\dfrac{5}{12}$. If $x$ represents the total time to finish the job, $\dfrac{5}{12}x=10\implies x=24$.
4 A container is $\dfrac{1}{4}$ full of water. After $12$ cups of water are added, it is $\dfrac{5}{8}$ full. How much water does the container hold? The portion of the container that 12 cups of water contributes is $\dfrac{5}{8}- \dfrac{1}{4}=\dfrac{3}{8}$. $\dfrac{3}{8}\cdot x=12\implies x=8\cdot \dfrac{12}{3}=32$ cups.
4 Sally is driving directly southeast at~30 mph. Which vector is closest to representing her travel? $\left < \dfrac{30}{\sqrt{2}},\dfrac{-30}{\sqrt{2}}\right > \implies 21\hat{\imath}-21\hat{\jmath}$
Find the total of what is inside the absolute value bars. If that is negative, take the positive or absolute value of it. Then simplify further.
1 , C What is $|5-x|$ when $x=17$? Substituting, $|5- 17|=|-12|=12$.
1 , C What is $|-27|- |12- |$?$27- |-32|=27- 32= -5$
1 , C , C What is $|11- 4|- |3- 8|$?$|7|- |-5|=7- 5=2$.
These are fairly rare on the exam and reasonably difficult. The key is to split them into two equations.
2 , C What are the solutions to $|3x- 5|=2$? Split the absolute value portion in 2. $3x- 5=2$ or $3x- 5=-2\implies 3x=7$ or $3x=3\implies x=\dfrac{7}{3}$ or $x=1$.
5 A difficult problem, which is generally not covered in school is what are the solutions to $|x|^2+3|x|- 10=0$? If you can graph this equation with your calculator including the absolute value and read the solutions, that may be the best approach. It does not matter whether the $x^2$ term has an absolute value around it or not, as it is positive anyway. You split this into 2 equations, $x^2+3x- 10=0$ and $x^2- 3x- 10=0$. The solutions to the first equation are $-5$ and $2$ and to the second equation $-2$ and 5. These can be obtained using the quadratic formula, factoring, by completing the square, or by graphing the expressions on your calculator. Only 2 and $-2$ check. You can also see that the first equation applies only for positive $x$ and the second one for negative values of $x$, so the $-5$ and 5 are extraneous.(Obtained values of $x$ not satisfying the equation are called extraneous roots)
There are some important issues with absolute value inequalities. You need to split them into separate cases, taking the negative of the expression inside the absolute value. If you take the negative of the number the absolute value is compared to, you will get a wrong answer. Also, when multiplying by a negative, you need to switch the direction of the inequality. However, it is usually possible to avoid multiplying by a negative.
3 What is the solution to $|x-5|< 2$?$x- 5 < 2$ and $5-x < 2\implies x < 7$ and $ x > 3 $, so solution is the interval $(3,7)$.
5T What is the solution to $|x-5|< -2$? Absolute value can never be negative, so there are no solutions. If you work it algebraically, you will get answers, but they do not check.
4 The solution for $|3x-a|< 2$ is $ \dfrac{11}{3}< x< 5$. What is $a$? $3x- a< 2$ and $a- 3x< 2\implies 3x< 2+a$ and $a- 2< 3x\implies x<\dfrac{2+a}{3}$ and $\dfrac{a- 2}{3}< x$. Therefore, $\dfrac{2+a}{3}=5\implies 2+a=15\implies a=13$. $13$ checks in the other equation.
$\sqrt{x^2} = x$, $\sqrt[3]{x^3} = x$, etc. Use that to simplify roots of variables and of constants in factored form.
4 What is $\sqrt[3]{80}$ simplified?$\sqrt[3]{2^4\cdot 5}=\sqrt[3]{2^3\cdot 2\cdot 5}=2 \sqrt[3]{10}$.
4 What is $5\sqrt[3]{3x^2}\sqrt[3]{2x^2}$? $5 \sqrt[3]{6x^4}=5x \sqrt[3]{6x}$.
4 What is $\sqrt[3]{\sqrt[5]{x^4}}$? $\sqrt[15]{x^4}$ or $x^{4/15}$.
4 What is $\left(\dfrac{9}{7}\right)^{-5/2}$?$\left(\dfrac{7}{9}\right)^{5/2}=\dfrac{7^{5/2}}{9^{5/2}}=\dfrac{49\sqrt{7}}{3^5}=\dfrac{49\sqrt{7}}{243}$.
4 What is $\left(\dfrac{8}{125}\right)^{-2/3}$? $\left(\dfrac{125}{8}\right)^{2/3}=\dfrac{5^2}{2^2}=\dfrac{25}{4}$.
Generally, simplify each term first. They will have the same squareroot term and can then be added.
4 , C What is $\sqrt{12}+\sqrt{75}- \sqrt{27}$? $2\sqrt{3}+5\sqrt{3}- 3\sqrt{3}=4\sqrt{3}$ or $\sqrt{48}$.
4 What is $\dfrac{12\sqrt{45}}{2\sqrt{5}}$? $\dfrac{36\sqrt{5}}{2\sqrt{5}}=18$.
Raise the expression to the power of the numerator and take the root indicated by the denominator.
4 What is $a^{1/2}b^{1/4}c^{5/8}$ in simplest radical form? $a^{4/8}b^{2/8}c^{5/8}=(a^4b^2c^5)^{1/8} = \sqrt[8]{a^4b^2c^5}$.
4 What is $x^{1/2}x^{1/6}$ is simplest radical form? $x^{3/6}x^{1/6}=x^{4/6}=x^{2/3}=\sqrt[3]{x^2}$.
Generally, isolate the radical first and then take both sides to the power of the radical to clear the radical.
Some problems can be solved by backsolving (plugging in and checking the answer choices). That is the preferred method for the SAT where some solutions generally do not check, so working the problem algebraically is a waste of time. However, for the ACT it is generally fine to work the problem algebraically.
3 $\sqrt{x}+\sqrt{4}=\sqrt{49}$. What is $x$?$\sqrt{x}+2=7\implies \sqrt{x}=5\implies x=25$.
4 $\sqrt[3]{x+3}+3=7$. What is $x$?$\sqrt[3]{x+3}=4\implies x+3=64\implies x=61$.
4 Solve $\sqrt{x+4}- 3>4$.$\sqrt{x+4}>7\implies x+4>49\implies x>45$.
Problems on the exam are generally best solved by solving for one of the variables in an equation and then substituting into the other equation. In some cases, you can add the equations and one of the variables will drop out. A more sophisticated approach involves linear combinations, but that is not needed for this exam. You can also graph the equations with your calculator and find the intersection point. There are programs for the calculator which will solve these problems. These problems can usually also be solved by plugging in answer choices.
3 , C If $x+y=17$ and $x-y=9$, what does $x$ equal? Adding the equations, $2x=26\implies x=13$.
4 What is the solution of $2x-3y=2$ and $x+5y=14$?The easiest approach is to use substitution and solve the second equation for $x$. $x=14-5y$. $2(14-5y)-3y=2\implies 28-10y-3y=2\implies 13y=26\implies y=2$. $x+5\cdot 2=14\implies x=4$, so $(4,2)$.
If two lines have different slopes, they intersect at one point. If the slopes are the same, they are usually parallel lines. If they are multiples of each other, they are the same line, and have infinitely many points in common.
4 $2x+3y=7$ and $5x+2y=23$. What does $8x+y$ equal? One way is to subtract the first equation from the second equation. $10x+4y=46$. Subtracting gives $8x+y=39$. Therefore, the answer is 39. You can also solve the system by substitution or linear combinations and then plug the solutions into $8x+y$.
4 Apples and pears have constant prices. The combination of 3 apples and 5 pears costs \$5.66, while 6 pears costs $\$4.02$. How much does each apple cost?
Each pear costs $\dfrac{4.02}{6}=0.67$. $3A+5\cdot 0.67=5.66\implies 3A+3.35=5.66\implies 3A=2.31\implies A=0.77$ or $77$ cents.
4 Sally has 101 marbles, all of which are red, blue, or green. She has 10 more blue marbles than red marbles, and 6 more green marbles than blue marbles. How many green marbles does she have? $R+B+G=101$, $B=R+10$, $G=B+6$. Therefore, $G=R+16$. $R+R+10+R+16=101$. $3R=75\implies R=25 \implies G=25+15=40$
It is easiest to do these problems using the formula $\dfrac {-b}{a}$.
3 ,F , C What is the sum of the solutions of $2x^2-10x+17=0$? You could find the solutions and add them, but it is easier to use the formula $\dfrac{-b}{a}$: $\dfrac{-(-10)}{2}=5$.
3 What is the nature of the solutions of the equation graphed? Two complex solutions, because the parabola does not cross the $x$-axis. If it crossed in two places, there would be two real solutions. If it just touched the $x$-axis, there would be exactly one real solution.
4 What is the nature of the solutions of $2x^2+11x+15=0$? Two real solutions. This can be determined by solving the equation by factoring. Also, if you use the quadratic formula, the discriminant (what is under the radical) is positive: $b^2- 4ac=11^2- 4\cdot 2\cdot 15=121-120=1$. If the discriminant was negative, there would be two complex solutions. If the discriminant was zero, there would be one real solution.
3 For what value of $k$ does $x^2-12x+k=0$ have exactly one solution? Completing the square, $\left(\dfrac{-12}{2}\right)^2=36$. Or set the discriminant equal to 0, $b^2- 4ac=0\implies (-12)^2-4\cdot 1\cdot k=0\implies 144=4k\implies k=36$.
Take $x$ minus each solution, FOIL out, and set equal to 0.
3 Which of the following is an equation with solutions $-5$ and 8? $(x+5)(x-8)=x^2-3x-40$.
4 Which of the following is an equation with solutions $\dfrac{2}{3}$ and $\dfrac{1}{5}$? $\left(x-\dfrac{2}{3}\right) \left(x-\dfrac{1}{5}\right)=0 \implies x^2-\dfrac{2x}{3}- \dfrac{x}{5}+\dfrac{2}{15}=0\implies x^2- \dfrac{13x}{15}+\dfrac{2}{15}=15x^2-13x+2=0$. Or (first multiplying by constants to clear the fractions) $(3x-2)(5x-1)=15x^2-3x- 10x+2=15x^2-13x+2=0$.
The quadratic formula is usually not provided, so it is helpful to memorize it: $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
4 , C , F What are the solutions of $x^2+5x=14$? Get all the terms on one side, $x^2+5x-14=0$. Now by factoring or using the quadratic formula, the solutions are $-7$ and 2. It is also possible to plug the solution choices in and see which works.
