2020 AMC 8 Solutions — Written & Video | LIVE by Po-Shen Loh

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2020 AMC 8 Solutions

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All problems are used with official legal permission of the Mathematical Association of America (MAA).

1.

Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?

$6$

$8$

$12$

$18$

$24$

Concepts:ratio and proportion

Difficulty rating: 370

Small Hint:

Water is $4\cdot2$ times the lemon juice

Big Hint:

First find the cups of sugar from the lemon juice

Video solution:

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Written solution:

Since Luka needs twice as much sugar as lemon, he needs $2\cdot3=6$ cups of sugar. Since Luka also needs $4$ times as much water as sugar, he needs $4\cdot6=24$ cups of water.

Thus, the correct answer is E.

2.

Four friends do yardwork for their neighbors over the weekend, earning $\$15,$ $\$20,$ $\$25,$ and $\$40$ respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $\$40$ give to the others?

$\$5$

$\$10$

$\$15$

$\$20$

$\$25$

Concepts:mean money

Difficulty rating: 450

Small Hint:

Find the equal share of the total earnings

Big Hint:

The $\$40$ earner gives away the amount above the equal share

Video solution:

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Written solution:

First, the total amount of money that they make is $\$15+\$20+\$25+\$40=\$100.$

Since they divide this equally, they each get $\frac{100}{4}=25$ dollars.

The friend who earned $\$40$ therefore gives away $\$40-\$25=\$15.$

Thus, the correct answer is C.

3.

Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest?

$560$

$960$

$1120$

$1920$

$3840$

Concepts:area rate

Difficulty rating: 560

Small Hint:

First find the garden area

Big Hint:

Multiply area by plants per square foot, then by strawberries per plant

Video solution:

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Written solution:

First, the size of the garden is $6\cdot8=48$ square feet.

Next, since there are $4$ plants per square foot, Carrie can plant $4\cdot 48=192$ plants total.

Finally, since there are $10$ strawberries per plant, Carrie can harvest $10\cdot 192=1920$ strawberries total.

Thus, the correct answer is D.

4.

Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?

$35$

$37$

$39$

$43$

$49$

Concepts:arithmetic sequence pattern recognition

Difficulty rating: 900

Small Hint:

The added bands have $6,$ $12,$ $18,$ $\ldots$ dots

Big Hint:

The next hexagon is the third one plus the next band

Video solution:

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Written solution:

The first three hexagons contain $1,$ $7,$ and $19$ dots. Each new band adds $6$ more dots than the previous band.

The fourth hexagon adds $18$ dots around the third hexagon, so it contains $19+18=37$ dots.

Thus, the correct answer is B.

5.

Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?

$5$

$10$

$15$

$20$

$25$

Concepts:fraction percentage

Difficulty rating: 720

Small Hint:

Each cup gets one fifth of $\frac{3}{4}$ of the pitcher

Big Hint:

Convert $\frac{3}{20}$ to a percent

Video solution:

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Written solution:

Each cup receives one fifth of the $\dfrac{3}{4}$ of a pitcher, or $\dfrac{1}{5}\cdot\dfrac{3}{4}=\dfrac{3}{20}=15\%.$

Thus, the correct answer is C.

6.

Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?

Aaron

Darren

Karen

Maren

Sharon

Concepts:logical deduction casework

Difficulty rating: 960

Small Hint:

Maren is fixed in the last car

Big Hint:

Test the possible adjacent positions for Sharon and Aaron

Video solution:

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Written solution:

Maren is in the last car. Since Aaron is directly behind Sharon and Darren is in front of Aaron, the possible placements of Darren, Sharon, and Aaron before Maren are limited.

If Sharon and Aaron were in cars $1$ and $2,$ Darren could not be in front of Aaron. If they were in cars $3$ and $4,$ then Darren and Karen would have to occupy cars $1$ and $2,$ which are adjacent.

Thus Sharon and Aaron must be in cars $2$ and $3.$ Darren is then in car $1,$ Karen is in car $4,$ and Aaron is in the middle car.

Thus, the correct answer is A.

7.

How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2357$ is one such integer.)

$9$

$10$

$15$

$21$

$28$

Concepts:digits combinations

Difficulty rating: 1020

Small Hint:

The first two digits must be $2$ and $3$

Big Hint:

Choose the last two digits from $4,$ $5,$ $6,$ $7,$ $8,$ $9$

Video solution:

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Written solution:

The thousands digit must be $2.$ Since the digits are distinct and increasing, the hundreds digit must be $3.$

The last two digits must be chosen from $4,$ $5,$ $6,$ $7,$ $8,$ $9.$ Once the two digits are chosen, their order is forced.

There are $\binom{6}{2}=15$ such integers.

Thus, the correct answer is C.

8.

Ricardo has $2020$ coins, some of which are pennies ( $1$ -cent coins) and the rest of which are nickels ( $5$ -cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?

$8062$

$8068$

$8072$

$8076$

$8082$

Concepts:linear equation optimization

Difficulty rating: 1020

Small Hint:

Leave one coin of each type fixed

Big Hint:

Changing one penny to one nickel changes the value by $4$ cents

Video solution:

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Written solution:

The greatest value occurs with $2019$ nickels and $1$ penny, while the least occurs with $1$ nickel and $2019$ pennies.

Between these two cases, $2018$ pennies have been replaced by nickels. Each replacement adds $5-1=4$ cents, so the difference is $2018\cdot4=8072$ cents.

Thus, the correct answer is C.

9.

Akash’s birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many small pieces will have icing on exactly two sides?

$12$

$16$

$18$

$20$

$24$

Concepts:cube geometry casework

Difficulty rating: 1100

Small Hint:

Separate the top layer from the other layers

Big Hint:

Top edge pieces and lower corner pieces are the exactly-two-sided pieces

Video solution:

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Written solution:

In the top layer, the non-corner edge pieces have icing on exactly two sides. There are $4$ edges with $2$ such pieces each, for $8$ pieces.

In each of the other $3$ layers, the only pieces with exactly two iced sides are the $4$ corner pieces. This adds $3\cdot4=12$ pieces.

The total is $8+12=20.$

Thus, the correct answer is D.

10.

Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?

$6$

$8$

$12$

$18$

$24$

Concepts:arrangements with restrictions complementary counting

Difficulty rating: 960

Small Hint:

Count all arrangements first

Big Hint:

Subtract arrangements where Steelie and Tiger are treated as one block

Video solution:

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Written solution:

There are $4!=24$ total arrangements of the marbles.

If the Steelie and Tiger are adjacent, treat them as one block. Then there are $3!$ ways to arrange the block with the other two marbles, and $2$ orders inside the block, for $3!\cdot2=12$ adjacent arrangements.

Therefore, $24-12=12$ arrangements keep the Steelie and Tiger separated.

Thus, the correct answer is C.

11.

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi’s and Maya’s average speeds?

$6$

$12$

$18$

$20$

$24$

Concepts:distance rate and time data and graph interpretation

Difficulty rating: 960

Small Hint:

Read each person’s time to travel $6$ miles

Big Hint:

Convert $10$ minutes and $30$ minutes to hours

Video solution:

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Written solution:

Naomi traveled $6$ miles in $10$ minutes, so her average speed was $\dfrac{6}{\frac{10}{60}}=36$ miles per hour.

Maya traveled $6$ miles in $30$ minutes, so her average speed was $\dfrac{6}{\frac{30}{60}}=12$ miles per hour.

This difference is $36-12=24.$

Thus, the correct answer is E.

12.

For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ For example:

$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$

What value of $N$ satisfies the following equation?

$5! \times 9! = 12 \times N!$

$10$

$11$

$12$

$13$

$14$

Concepts:factorial

Difficulty rating: 1020

Small Hint:

Cancel the factor $12$ from $5!$

Big Hint:

Rewrite $5!$ so the factor $12$ cancels cleanly

Video solution:

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Written solution:

Because $5!=120=12\cdot10,$ the equation becomes $12\cdot10\cdot9!=12\cdot N!.$ Canceling $12$ gives $N!=10\cdot9!=10!,$ so $N=10.$

Thus, the correct answer is A.

13.

Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

$6$

$9$

$12$

$18$

$24$

Concepts:basic probability linear equation

Difficulty rating: 1020

Small Hint:

The non-purple socks stay at $18$

Big Hint:

If purple is $60\%$ , non-purple is $40\%$

Video solution:

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Written solution:

Suppose Jamal adds $s$ purple socks. Then, there will be $s+18$ purple socks.

Also, since there are $36$ total socks to begin with, we have $s+36$ socks after adding the socks.

Since we have a $60\%$ chance of choosing a purple sock afterwards, we know $\dfrac{s+18}{s+36}=0.6.$

Solving for $s$ yields: $\begin{align*}s+18 &= 0.6s+21.6 \\ 0.4s &= 3.6\\ s &= 9.\end{align*}$

Therefore, $9$ socks are added.

Thus, the correct answer is B.

14.

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities?

$65{,}000$

$75{,}000$

$85{,}000$

$95{,}000$

$105{,}000$

Concepts:mean data and graph interpretation

Difficulty rating: 870

Small Hint:

The dashed line is just under $5000$

Big Hint:

Total population = average population times $20$

Video solution:

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Written solution:

Looking at the horizontal dashed line, the average population is around $4750.$

Since there are $20$ cities, the total population is approximately $20\cdot4750=95{,}000.$

Therefore, the total population is approximately $95{,}000.$

Thus, the correct answer is D.

15.

Suppose $15\%$ of $x$ equals $20\%$ of $y.$ What percentage of $x$ is $y?$

$5$

$35$

$75$

$133 \frac{1}{3}$

$300$

Concepts:percentage ratio and proportion

Difficulty rating: 960

Small Hint:

Translate the sentence as $0.15x=0.20y$

Big Hint:

Solve for $\frac{y}{x}$

Video solution:

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Written solution:

The statement gives $0.15x=0.20y.$

Solving for $y$ gives $y=\dfrac{0.15}{0.20}x=\dfrac{3}{4}x.$

Therefore, $y$ is $75\%$ of $x.$

Thus, the correct answer is C.

16.

Each of the points $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$

$1$

$2$

$3$

$4$

$5$

Concepts:double counting summation

Difficulty rating: 1270

Small Hint:

Most points are counted twice in the five line sums

Big Hint:

Point $B$ is counted one extra time

Video solution:

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Written solution:

Every point is counted in two of the line sums except $B,$ which is counted in three. Thus the total of the five line sums is twice the sum of all six digits, plus one extra $B.$

The digits are exactly $1,$ $2,$ $3,$ $4,$ $5,$ and $6,$ whose sum is $21.$ Therefore $2\cdot21+B=47,$ so $B=5.$

Thus, the correct answer is E.

17.

How many factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,$ $2,$ $3,$ $4,$ $6,$ and $12.$ )

$6$

$7$

$8$

$9$

$10$

Concepts:factor counting prime factorization

Difficulty rating: 1190

Small Hint:

List or factor the divisors of $2020$

Big Hint:

Only $1$ , primes, and $4$ have at most $3$ factors

Video solution:

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Written solution:

Since $2020=2^2\cdot5\cdot101,$ it has $(2+1)(1+1)(1+1)=12$ positive factors.

Among those factors, only $1$ and the primes $2,$ $5,$ and $101$ have fewer than three factors, while $4$ has exactly three. The other $12-5=7$ factors therefore have more than three factors.

Thus, the correct answer is B.

18.

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$

$240$

$248$

$256$

$264$

$272$

Concepts:circle Pythagorean Theorem rectangle

Difficulty rating: 1330

Small Hint:

The diameter is $9+16+9$

Big Hint:

Use the radius and half the rectangle base in a right triangle

Video solution:

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Written solution:

Since $FE$ is the diameter of the semicircle, we know the length of the diameter is $34,$ and so the radius is $17.$ Let $O$ be the center of the diameter.

The length from $OF$ therefore is $17.$

Since $D$ is on $OF,$ we know $\begin{align*} OD + FD &= OF\\ OD + 9 &= 17 \\ OD &= 8. \end{align*}$

Also, since we have a semicircle, we know $OC = 17.$

Finally, since $ABCD$ is a rectangle, we know $\angle ODC$ is a right angle. This means we can find $DC$ by the Pythagorean Theorem. We know $\begin{align*} OD^2+DC^2&=OC^2 \\ 8^2+DC^2 &= 17^2 \\ DC &= 15. \end{align*}$

Thus, the area of the rectangle is $DC\cdot DA = 15\cdot 16=240.$

Thus, the correct answer is A.

19.

A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$

$3$

$4$

$5$

$6$

$8$

Concepts:divisibility digits

Difficulty rating: 1270

Small Hint:

A five-digit flippy number has form $ABABA$

Big Hint:

Divisibility by $5$ forces $A=5$

Video solution:

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Written solution:

A five-digit flippy number has the form $ABABA,$ where $A\ne0$ and $A\ne B.$ Divisibility by $5$ requires the last digit $A$ to be $0$ or $5.$ Since $A\ne0,$ we must have $A=5.$

The number therefore has the form $5B5B5.$ Its digit sum is $15+2B,$ so divisibility by $3$ requires $B$ to be a multiple of $3.$ The possibilities are $B=0,$ $B=3,$ $B=6,$ and $B=9,$ giving $4$ numbers.

Thus, the correct answer is B.

20.

A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

$\begin{array}{|c|c|}\hline T_1&\underline{\phantom{00}}\,\mathrm m\\T_2&11\,\mathrm m\\T_3&\underline{\phantom{00}}\,\mathrm m\\T_4&\underline{\phantom{00}}\,\mathrm m\\T_5&\underline{\phantom{00}}\,\mathrm m\\\hline\overline T&\underline{\phantom{00}}.2\,\mathrm m\\\hline\end{array}$

$22.2$

$24.2$

$33.2$

$35.2$

$37.2$

Concepts:casework mean

Difficulty rating: 1370

Small Hint:

Integer heights force the neighbors of $11$ to be $22$

Big Hint:

The average ending in $0.2$ means the total ends in $1$ mod $5$

Video solution:

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Written solution:

Tree $2$ is $11$ meters tall. Since all heights are integers and neighboring trees differ by a factor of $2,$ trees $1$ and $3$ must both be $22$ meters tall.

The first three trees total $22+11+22=55$ meters. The possible integer pairs for trees $4$ and $5$ are $(11,22),$ $(44,22),$ and $(44,88).$

These give averages $17.6,$ $24.2,$ and $37.4,$ respectively. The notebook shows that the average ends in $0.2,$ so it must be $24.2.$

Thus, the correct answer is B.

21.

A game board consists of $64$ squares that alternate between shaded and unshaded. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining unshaded squares in the row above. How many $7$ -step paths are there from $P$ to $Q?$ (The figure shows a sample path.)

$28$

$30$

$32$

$33$

$35$

Concepts:lattice paths Pascal’s Triangle

Difficulty rating: 1460

Small Hint:

Each move goes one row up and one column left or right

Big Hint:

Add path counts from the two adjoining squares below

Video solution:

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Written solution:

Each move must go up one row and either left or right. Counting row by row from $P,$ each reachable square gets the sum of the counts from the two adjoining squares below it.

The count at $Q$ is $28,$ so there are $28$ paths.

Thus, the correct answer is A.

22.

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 + 1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ $\begin{align*} &7 \to 22 \to 11 \to 34 \\ &\to 17 \to 52 \to 26 \end{align*}$ When the same $6$ -step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ $\begin{align*} &N \to \underline{\phantom{00}} \to \underline{\phantom{00}} \to \underline{\phantom{00}}\\ &\to \underline{\phantom{00}} \to \underline{\phantom{00}} \to 1 \end{align*}$

$73$

$74$

$75$

$82$

$83$

Concepts:work backwards tree diagram

Difficulty rating: 1670

Small Hint:

Work backward from the final output $1$

Big Hint:

A previous value can be $2m$ , and sometimes $\frac{m-1}{3}$

Video solution:

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Written solution:

Work backward from $1.$ A value $m$ can always come from the even input $2m.$ It can also come from the odd input $\frac{m-1}{3}$ when that expression is a positive odd integer.

Listing the possible values after each backward step gives $\begin{aligned} \{1\}&\leftarrow\{2\}\leftarrow\{4\}\leftarrow\{1,8\}\\ &\leftarrow\{2,16\}\leftarrow\{4,5,32\}\\ &\leftarrow\{1,8,10,64\}. \end{aligned}$ Thus the possible starting values are $1,$ $8,$ $10,$ $64,$ whose sum is $83.$

Thus, the correct answer is E.

23.

Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?

$120$

$150$

$180$

$210$

$240$

Concepts:inclusion-exclusion multiplication principle

Difficulty rating: 1370

Small Hint:

Start with $3^5$ unrestricted distributions

Big Hint:

Use inclusion-exclusion for students receiving no awards

Video solution:

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Written solution:

There are $3^5=243$ ways to give each of the $5$ distinct awards to one of the $3$ students.

Subtract the distributions in which at least one student receives no award. If a particular student receives none, the awards go to the other two students in $2^5$ ways. This gives $3\cdot2^5$ counts, but the $3$ cases in which one student receives all awards have each been subtracted twice.

By inclusion-exclusion, the desired number is $3^5-3\cdot2^5+3=150.$

Thus, the correct answer is B.

24.

A large square region is paved with $n^2$ shaded square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n=3.$ When $n=24,$ the $576$ shaded tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n?$

$\dfrac{6}{25}$

$\dfrac{1}{4}$

$\dfrac{9}{25}$

$\dfrac{7}{16}$

$\dfrac{9}{16}$

Concepts:area ratio square (geometry)

Difficulty rating: 1510

Small Hint:

The large side length is $24s+25d$

Big Hint:

Compare shaded area $24^2s^2$ with total area

Video solution:

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Written solution:

For $n=24,$ the shaded tile area is $24^2s^2.$ Each side of the large square consists of $24$ tiles and $25$ borders, so its side length is $24s+25d.$

The shaded tiles cover $64\%=\dfrac{16}{25}$ of the large square, so $\dfrac{24^2s^2}{(24s+25d)^2}=\dfrac{16}{25}.$ Taking positive square roots gives $\dfrac{24s}{24s+25d}=\dfrac{4}{5}.$

Thus $120s=96s+100d,$ so $24s=100d$ and $\dfrac{d}{s}=\dfrac{6}{25}.$

Thus, the correct answer is A.

25.

Rectangles $R_1$ and $R_2,$ and squares $S_1,$ $S_2,$ and $S_3,$ shown below, combine to form a rectangle that is $3322$ units wide and $2020$ units high. What is the side length of $S_2$ in units?

$651$

$655$

$656$

$662$

$666$

Concepts:system of equations rectangle

Difficulty rating: 1370

Small Hint:

Let the square side lengths be $s_1,s_2,s_3$

Big Hint:

Subtract the height equation from the width equation

Video solution:

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Written solution:

Let $s_1,s_2,s_3$ be the side lengths of the three squares. Across the width of the large rectangle, $s_1+s_2+s_3=3322.$

The height of $R_2$ is $s_1-s_2,$ so the total height is $s_1-s_2+s_3=2020.$ Subtracting the height equation from the width equation gives $2s_2=1302,$ so $s_2=651.$

Thus, the correct answer is A.