Found problems: 260
1999 AMC 8, 18
Problems 17, 18, and 19 refer to the following:
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.
They learn that a big concert is scheduled for the same night and attendance will be down $25\%$. How many recipes of cookies should they make for their smaller party?
$ \text{(A)}\ 6\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11 $
2007 AMC 8, 1
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work during the final week to earn the tickets?
$\textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 10 \qquad
\textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 13$
2017 AMC 8, 10
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
$\textbf{(A) }\frac{1}{10}\qquad\textbf{(B) }\frac{1}{5}\qquad\textbf{(C) }\frac{3}{10}\qquad\textbf{(D) }\frac{2}{5}\qquad\textbf{(E) }\frac{1}{2}$
2023 AMC 8, 10
Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?
$\textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{5}{12}$
2020 AMC 8 -, 7
How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2357$ is one such integer.)
$\textbf{(A) }9 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }21 \qquad \textbf{(E) }28$
2023 AMC 8, 19
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\tfrac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
[asy]
size(5cm);
fill((0,0)--(2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(4,0)--cycle,lightgray*0.5+mediumgray*0.5);
draw((0,0)--(4,0)--(2,2*sqrt(3))--cycle);
//center: 2,1.155
draw((2/3,1.155/3)--(4-(4-2)/3,1.155/3)--(2,2*sqrt(3)-0.7697)--cycle);
dot((0,0)^^(4,0)^^(2,2*sqrt(3))^^(2/3,1.155/3)^^(4-(4-2)/3,1.155/3)^^(2,2*sqrt(3)-0.7697));
draw((0,0)--(2/3,1.155/3));
draw((4,0)--(4-(4-2)/3,1.155/3));
draw((2,2*sqrt(3))--(2,2*sqrt(3)-0.7697));
[/asy]
$\textbf{(A) } 1:3\qquad\textbf{(B) } 3:8\qquad\textbf{(C) } 5:12\qquad\textbf{(D) } 7:16\qquad\textbf{(E) } 4:9$
2020 AMC 8 -, 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?
$\textbf{(A) }8062 \qquad \textbf{(B) }8068 \qquad \textbf{(C) }8072 \qquad \textbf{(D) }8076 \qquad \textbf{(E) }8082$
2018 AMC 8, 17
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is $2$ miles, which is $10,560$ feet, and Bella covers $2 \tfrac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella?
$\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520$
2023 AMC 8, 7
A rectangle, with sides parallel to the $x-$axis and $y-$axis, has opposite vertices located at $(15, 3)$ and$(16, 5).$ A line is drawn through points $A(0, 0)$ and $B(3, 1).$ Another line is drawn through points $C(0, 10)$ and $D(2, 9).$ How many points on the rectangle lie on at least one of the two lines?
[asy]
size(9cm);
draw((0,-.5)--(0,11),EndArrow(size=.15cm));
draw((1,0)--(1,11),mediumgray);
draw((2,0)--(2,11),mediumgray);
draw((3,0)--(3,11),mediumgray);
draw((4,0)--(4,11),mediumgray);
draw((5,0)--(5,11),mediumgray);
draw((6,0)--(6,11),mediumgray);
draw((7,0)--(7,11),mediumgray);
draw((8,0)--(8,11),mediumgray);
draw((9,0)--(9,11),mediumgray);
draw((10,0)--(10,11),mediumgray);
draw((11,0)--(11,11),mediumgray);
draw((12,0)--(12,11),mediumgray);
draw((13,0)--(13,11),mediumgray);
draw((14,0)--(14,11),mediumgray);
draw((15,0)--(15,11),mediumgray);
draw((16,0)--(16,11),mediumgray);
draw((-.5,0)--(17,0),EndArrow(size=.15cm));
draw((0,1)--(17,1),mediumgray);
draw((0,2)--(17,2),mediumgray);
draw((0,3)--(17,3),mediumgray);
draw((0,4)--(17,4),mediumgray);
draw((0,5)--(17,5),mediumgray);
draw((0,6)--(17,6),mediumgray);
draw((0,7)--(17,7),mediumgray);
draw((0,8)--(17,8),mediumgray);
draw((0,9)--(17,9),mediumgray);
draw((0,10)--(17,10),mediumgray);
draw((-.13,1)--(.13,1));
draw((-.13,2)--(.13,2));
draw((-.13,3)--(.13,3));
draw((-.13,4)--(.13,4));
draw((-.13,5)--(.13,5));
draw((-.13,6)--(.13,6));
draw((-.13,7)--(.13,7));
draw((-.13,8)--(.13,8));
draw((-.13,9)--(.13,9));
draw((-.13,10)--(.13,10));
draw((1,-.13)--(1,.13));
draw((2,-.13)--(2,.13));
draw((3,-.13)--(3,.13));
draw((4,-.13)--(4,.13));
draw((5,-.13)--(5,.13));
draw((6,-.13)--(6,.13));
draw((7,-.13)--(7,.13));
draw((8,-.13)--(8,.13));
draw((9,-.13)--(9,.13));
draw((10,-.13)--(10,.13));
draw((11,-.13)--(11,.13));
draw((12,-.13)--(12,.13));
draw((13,-.13)--(13,.13));
draw((14,-.13)--(14,.13));
draw((15,-.13)--(15,.13));
draw((16,-.13)--(16,.13));
label(scale(.7)*"$1$", (1,-.13), S);
label(scale(.7)*"$2$", (2,-.13), S);
label(scale(.7)*"$3$", (3,-.13), S);
label(scale(.7)*"$4$", (4,-.13), S);
label(scale(.7)*"$5$", (5,-.13), S);
label(scale(.7)*"$6$", (6,-.13), S);
label(scale(.7)*"$7$", (7,-.13), S);
label(scale(.7)*"$8$", (8,-.13), S);
label(scale(.7)*"$9$", (9,-.13), S);
label(scale(.7)*"$10$", (10,-.13), S);
label(scale(.7)*"$11$", (11,-.13), S);
label(scale(.7)*"$12$", (12,-.13), S);
label(scale(.7)*"$13$", (13,-.13), S);
label(scale(.7)*"$14$", (14,-.13), S);
label(scale(.7)*"$15$", (15,-.13), S);
label(scale(.7)*"$16$", (16,-.13), S);
label(scale(.7)*"$1$", (-.13,1), W);
label(scale(.7)*"$2$", (-.13,2), W);
label(scale(.7)*"$3$", (-.13,3), W);
label(scale(.7)*"$4$", (-.13,4), W);
label(scale(.7)*"$5$", (-.13,5), W);
label(scale(.7)*"$6$", (-.13,6), W);
label(scale(.7)*"$7$", (-.13,7), W);
label(scale(.7)*"$8$", (-.13,8), W);
label(scale(.7)*"$9$", (-.13,9), W);
label(scale(.7)*"$10$", (-.13,10), W);
dot((0,0));
label(scale(.65)*"$A$", (0,0), NE);
dot((3,1));
label(scale(.65)*"$B$", (3,1), NE);
dot((0,10));
label(scale(.65)*"$C$", (0,10), NE);
dot((2,9));
label(scale(.65)*"$D$", (2,9), NE);
draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125));
dot((15,3));
dot((16,3));
dot((16,5));
dot((15,5));
[/asy]
$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$
2016 AMC 8, 1
The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?
$\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$
2024 AMC 8 -, 13
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$
2016 AMC 8, 11
Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is $132.$
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }12$
2019 AMC 8, 23
After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?
$\textbf{(A) }10\qquad\textbf{(B) }11\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }14$
2017 AMC 8, 5
What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$?
$\textbf{(A) }1020\qquad\textbf{(B) }1120\qquad\textbf{(C) }1220\qquad\textbf{(D) }2240\qquad\textbf{(E) }3360$
2010 AMC 8, 2
If $a @ b = \frac{a\times b}{a+b}$, for $a,b$ positive integers, then what is $5 @10$?
$\textbf{(A)}\ \frac{3}{10} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{10}{3} \qquad\textbf{(E)}\ 50$
2012 AMC 8, 9
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?
$\textbf{(A)}\hspace{.05in}61 \qquad \textbf{(B)}\hspace{.05in}122 \qquad \textbf{(C)}\hspace{.05in}139 \qquad \textbf{(D)}\hspace{.05in}150 \qquad \textbf{(E)}\hspace{.05in}161 $
2025 AMC 8, 21
The Konigsberg School has assigned grades $1$ through $7$ to pods $A$ through $G$, one grade per pod. The school noticed that each pair of connected pods has been assigned grades differing by $2$ or more grade levels. (For example, grades $1$ and $2$ will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E,$ and $F$?
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 16$\\
2002 AMC 12/AHSME, 25
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
[asy]//Choice A
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125;
}
real g(real x) {
return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(A)}$", (-5,4.5));
[/asy]
[asy]//Choice B
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2;
}
real g(real x) {
return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(B)}$", (-5,4.5));
[/asy]
[asy]//Choice C
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.21875 x^2+0.28125 x+0.5;
}
real g(real x) {
return -0.375 x^2-0.75 x+0.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(C)}$", (-5,4.5));
[/asy]
[asy]//Choice D
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875;
}
real g(real x) {
return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625;
}
real z=3.14;
draw(graph(f,-z, z), heavygray);
draw(graph(g,-z, z), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(D)}$", (-5,4.5));
[/asy]
[asy]//Choice E
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598;
}
real g(real x) {
return -0.166667 x^3+0.125 x^2+0.479167 x-0.375;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(E)}$", (-5,4.5));
[/asy]
2019 AMC 8, 6
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
[asy]
size(130);
defaultpen(fontsize(11));
int i, j;
for(i=0; i<9; i=i+1)
{
for(j=0; j<9; j=j+1)
if((i==4) && (j==4))
{
dot((i,j),linewidth(5));
} else {
dot((i,j),linewidth(3));
}
}
dot("$P$",(4,4),NE);
draw((0,0)--(0,8)--(8,8)--(8,0)--cycle);
[/asy]
$\textbf{(A) } \frac{1}{5}
\qquad\textbf{(B) } \frac{1}{4}
\qquad\textbf{(C) } \frac{2}{5}
\qquad\textbf{(D) } \frac{9}{20}
\qquad\textbf{(E) } \frac{1}{2}$
2020 AMC 8 -, 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?
$\textbf{(A) }\text{Aaron}\qquad \textbf{(B) }\text{Darren}\qquad \textbf{(C) }\text{Karen}\qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}\qquad$
2003 AMC 8, 18
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?
[asy]/* AMC8 2003 #18 Problem */
pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);
pair e=(269,105), f=(359,104), g=(303,12), h=(579,211);
pair i=(534, 342), j=(442,432), k=(374,484), l=(278,501);
pair m=(282,411), n=(147,451), o=(103,437), p=(31,373);
pair q=(419,175), r=(462,209), s=(477,288), t=(443,358);
pair oval=(282,303);
draw(l--m--n--cycle);
draw(p--oval);
draw(o--oval);
draw(b--d--oval);
draw(c--d--e--oval);
draw(e--f--g--h--i--j--oval);
draw(k--oval);
draw(q--oval);
draw(s--oval);
draw(r--s--t--oval);
dot(a); dot(b); dot(c); dot(d); dot(e); dot(f); dot(g); dot(h);
dot(i); dot(j); dot(k); dot(l); dot(m); dot(n); dot(o); dot(p);
dot(q); dot(r); dot(s); dot(t);
filldraw(yscale(.5)*Circle((282,606),80),white,black);
label(scale(0.75)*"Sarah", oval);[/asy]
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
2015 AMC 8, 7
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$
2002 AMC 8, 20
The area of triangle $ XYZ$ is 8 square inches. Points $ A$ and $ B$ are midpoints of congruent segments $ \overline{XY}$ and $ \overline{XZ}$. Altitude $ \overline{XC}$ bisects $ \overline{YZ}$. What is the area (in square inches) of the shaded region?
[asy]/* AMC8 2002 #20 Problem */
draw((0,0)--(10,0)--(5,4)--cycle);
draw((2.5,2)--(7.5,2));
draw((5,4)--(5,0));
fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey);
label(scale(0.8)*"$X$", (5,4), N);
label(scale(0.8)*"$Y$", (0,0), W);
label(scale(0.8)*"$Z$", (10,0), E);
label(scale(0.8)*"$A$", (2.5,2.2), W);
label(scale(0.8)*"$B$", (7.5,2.2), E);
label(scale(0.8)*"$C$", (5,0), S);
fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);[/asy]
$ \textbf{(A)}\ 1\frac12\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2\frac12\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 3\frac12$
2018 AMC 8, 6
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
$\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$
2019 AMC 8, 3
Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest?
$\textbf{(A) } \frac{15}{11} < \frac{17}{13} < \frac{19}{15} \qquad\textbf{(B) } \frac{15}{11} < \frac{19}{15} < \frac{17}{13} \qquad\textbf{(C) } \frac{17}{13} < \frac{19}{15} < \frac{15}{11}
\newline\newline
\qquad\textbf{(D) } \frac{19}{15} < \frac{15}{11} < \frac{17}{13} \qquad\textbf{(E) } \frac{19}{15} < \frac{17}{13} < \frac{15}{11}$