2005 CEMC Gauss (Grade 7) Problems

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Part A: Each correct answer is worth 5 points

Problem 1

The value of $\frac{3 \times 4}{6}$ is

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 6$

Solution

Problem 2

The value of $0.8 - 0.07$ is

$\text{(A)}\ 0.1 \qquad \text{(B)}\ 0.71 \qquad \text{(C)}\ 0.793 \qquad \text{(D)}\ 0.01 \qquad \text{(E)}\ 0.73$

Solution

Problem 3

[asy] //make the picture small and square size(80,80);  //draw an arc from 0 to 75 degrees of radius 1 centered at the origin draw(arc((0,0),1,0,75));  //draw a line with an arrow on the end pointing to 25 degrees (scale it down by .95 so that it stays inside the arc) //dir(25) is a unit vector pointing 25 degrees draw((0,0)--scale(.95)*dir(25),Arrow);  //put a label at the end of the 75 degree unit vector (and position it out 75 degrees) MarkPoint("9",dir(75),dir(75));  //put a label at the end of the 75 degree unit vector (rotate label 75 degrees to create a tick) MarkPoint(75,"-",scale(.85)*dir(75),dir(75)); MarkPoint("9.2",dir(60),dir(60)); MarkPoint(60,"-",scale(.85)*dir(60),dir(60)); MarkPoint("9.4",dir(45),dir(45));  //put a label at the end of the 45 degree unit vector - use NE to move it slightly NorthEast of this point (or use dir(45)) MarkPoint(45,"-",scale(.85)*dir(45),NE); MarkPoint("9.6",dir(30),dir(30)); MarkPoint(30,"-",scale(.85)*dir(30),dir(30)); MarkPoint("9.8",dir(15),dir(15)); MarkPoint(15,"-",scale(.85)*dir(15),dir(15));  //put a label at the end of the 0 degree unit vector - use E to move it slightly East of this point (or use dir(0)) MarkPoint("10",dir(0),E); MarkPoint(0,"-",scale(.85)*dir(0),dir(0)); [/asy]


Contestants on "Gauss Reality TV" are rated by an applause metre. In the diagram, the arrow for one of the contestants is pointing to a rating closest to:

$\text{(A)}\ 9.4 \qquad \text{(B)}\ 9.3 \qquad \text{(C)}\ 9.7 \qquad \text{(D)}\ 9.9 \qquad \text{(E)}\ 9.5$

Solution

Problem 4

Twelve million added to twelve thousand equals

$\text{(A)}\ 12,012,000 \qquad \text{(B)}\ 12,120,000 \qquad \text{(C)}\ 120,120,000 \qquad \text{(D)}\ 12,000,012,000 \qquad \text{(E)}\ 12,012,000,000$

Solution

Problem 5

The largest number in the set {$0.109, 0.2, 0.111, 0.114, 0.19$} is

$\text{(A)}\ 0.109 \qquad \text{(B)}\ 0.2 \qquad \text{(C)}\ 0.111 \qquad \text{(D)}\ 0.114 \qquad \text{(E)}\ 0.19$

Solution

Problem 6

At a class party, each student randomly selects a wrapped prize from a bag. The prizes include books and calculators. There are $27$ prizes in the bag. Meghan is the first to choose a prize. If the probability of Meghan choosing a book for her prize is $2/3$, how many books are in the bag?

$\text{(A)}\ 15 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 21 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 18$

Solution

Problem 7

Karen has just been chosen the new “Math Idol”. A total of $1,480,000$ votes were cast and Karen received $83\%$ of them. How many people voted for her?

$\text{(A)}\ 830,000 \qquad \text{(B)}\ 1,228,400 \qquad \text{(C)}\ 1,100,000 \qquad \text{(D)}\ 251,600 \qquad \text{(E)}\ 1,783,132$

Solution

Problem 8

In the diagram, what is the measure of $\angle ACB$ in degrees? [asy] size(300); draw((-60,0)--(0,0)); draw((0,0)--(64.3,76.6)--(166,0)--cycle); label("$A$",(64.3,76.6),N); label("$93^\circ$",(64.3,73),S); label("$130^\circ$",(0,0),NW); label("$B$",(0,0),S); label("$D$",(-60,0),S); label("$C$",(166,0),S); [/asy]

$\text{(A)}\ 57^\circ \qquad \text{(B)}\ 37^\circ \qquad \text{(C)}\ 47^\circ \qquad \text{(D)}\ 60^\circ \qquad \text{(E)}\ 17^\circ$

Solution

Problem 9

A movie theatre has eleven rows of seats. The rows are numbered from $1$ to $11$. Odd-numbered rows have $15$ seats and even-numbered rows have $16$ seats. How many seats are there in the theatre?

$\text{(A)}\ 176 \qquad \text{(B)}\ 186 \qquad \text{(C)}\ 165 \qquad \text{(D)}\ 170 \qquad \text{(E)}\ 171$

Solution

Problem 10

In relation to Smiths Falls, Ontario, the local time in St. John’s, Newfoundland, is $90$ minutes ahead, and the local time in Whitehorse, Yukon, is $3$ hours behind. When the local time in St. John’s is 5:36 p.m., the local time in Whitehorse is

$\text{(A)}$ 1:06 p.m. $\text{(B)}$ 2:36 p.m. $\text{(C)}$ 4:06 p.m. $\text{(D)}$ 12:06 p.m. $\text{(E)}$ 10:06 p.m.

Solution

Part B: Each correct answer is worth 6 points

Problem 11

The temperature range on a given day is the difference between the daily high and the daily low temperatures. On the graph shown, which day has the greatest temperature range?

$\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Tuesday} \qquad \text{(C)}\ \text{Wednesday} \qquad \text{(D)}\ \text{Thursday} \qquad \text{(E)}\ \text{Friday}$

[asy] size(300); draw((0,-10)--(0,10),EndArrow); // Comments this line out for reading purposes //draw((0,0)--(16,0),EndArrow); draw((0,0)--(16,0),arrow=Arrow(TeXHead));//smaller arrowhead picture temp; label(temp,"Temperature ($^\circ$C)"); add(rotate(90)*temp,(-2,0)); picture mon; label(mon,"Mon."); add(rotate(90)*mon,(2,7.3)); picture tues; label(tues,"Tues."); add(rotate(90)*tues,(4,4.3)); picture Wed; label(Wed,"Wed."); add(rotate(90)*Wed,(6,5.3)); picture Thurs; label(Thurs,"Thurs."); add(rotate(90)*Thurs,(8,5.4));  //picture Fri; //label(Fri,"Fri."); //add(rotate(90)*Fri,(10,9)); MP(90,"\textup{ Fri.}",(10,8),N);//MarkPoint(Rotation,Label,Coordinates) textup(upright no-italics) draw((-0.3,-8)--(0.3,-8)); draw((-0.3,-6)--(0.3,-6)); draw((-0.3,-4)--(0.3,-4)); draw((-0.3,-2)--(0.3,-2)); draw((-0.3,2)--(0.3,2)); draw((-0.3,4)--(0.3,4)); //draw((-0.3,6)--(0.3,6)); //draw((-0.3,8)--(0.3,8)); path Tick=((-.3,0)--(.3,0));draw(shift(0,6)*Tick);draw(shift(0,8)*Tick); dot((2,6)); dot((4,3)); dot((6,4)); dot((8,4)); dot((10,8)); draw((1.8,-3.8)--(2.2,-3.8)--(2.2,-4.2)--(1.8,-4.2)--cycle); draw((3.8,-5.8)--(4.2,-5.8)--(4.2,-6.2)--(3.8,-6.2)--cycle); draw((5.8,-1.8)--(6.2,-1.8)--(6.2,-2.2)--(5.8,-2.2)--cycle); draw((7.8,-4.8)--(8.2,-4.8)--(8.2,-5.2)--(7.8,-5.2)--cycle); //draw((9.8,0.2)--(10.2,0.2)--(10.2,-0.2)--(9.8,-0.2)--cycle); path Square=((-.2,-.2)--(-.2,.2)--(.2,.2)--(.2,-.2)--cycle); filldraw(shift(10,0)*Square,white);// this hides the x-axis behind white filling -- try green label("-8",(-0.3,-8),W); label("-6",(-0.3,-6),W); label("-4",(-0.3,-4),W); label("-2",(-0.3,-2),W); label("0",(-0.3,0),W); label("2",(-0.3,2),W); label("4",(-0.3,4),W); label("6",(-0.3,6),W); label("8",(-0.3,8),W); dot((12,-6.5)); label("Daily High",(12,-6.5),E); draw((11.8,-7.7)--(11.8,-7.3)--(12.2,-7.3)--(12.2,-7.7)--cycle); label("Daily Low",(12,-7.5),E); [/asy]

Solution

Problem 12

A bamboo plant grows at a rate of $105$ cm per day. On May 1st at noon it was $2 m$ tall. Approximately how tall, in metres, was it on May 8th at noon?

$\text{(A)}\ 10.40 \qquad \text{(B)}\ 8.30 \qquad \text{(C)}\ 3.05 \qquad \text{(D)}\ 7.35 \qquad \text{(E)}\ 9.35$

Solution

Problem 13

In the diagram, the length of $DC$ is twice the length of $BD$. What is the area of the triangle $ABC$?

$\text{(A)}\ 24 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 36$

[asy] draw((0,0)--(-3,0)--(0,4)--cycle); draw((0,0)--(6,0)--(0,4)--cycle); label("3",(-1.5,0),N); label("4",(0,2),E); label("$A$",(0,4),N); label("$B$",(-3,0),S); label("$C$",(6,0),S); label("$D$",(0,0),S); draw((0,0.4)--(0.4,0.4)--(0.4,0)); [/asy]

Solution

Problem 14

The numbers on opposite sides of a die total $7$. What is the sum of the numbers on the unseen faces of the two dice shown?

$\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 21 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 30$

[asy] import three; unitsize(1cm); size(100); currentprojection=orthographic(1/2,-1,1/2); // three - currentprojection, orthographic draw((0,0,0)--(0,0,1)); draw((1,1,0)--(1,1,1)); draw((0,0,0)--(1,0,0)); draw((1,1,0)--(1,0,0)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); dot((0,.5,0)); dot((0,1,0)); dot((0,1.5,0)); dot((-.5,1.5,0)); dot((0,1.5,-.5)); dot((1,.2,.3)); dot((1,.2,.7)); dot((1,.8,.3)); dot((1,.8,.7)); dot((.1,1.3,.6)); draw((-1,0,1)--(-1,0,0)--(-2,0,0)--(-2,0,1)); draw((-1,0,0)--(-1,1,0)); draw((-1,1,1)--(-1,1,0)); draw((-1,0,1)--(-2,0,1)--(-2,1,1)--(-1,1,1)--cycle); dot((-1.8,0,0.2)); dot((-1.5,0,0.2)); dot((-1.2,0,0.2)); dot((-1.8,0,0.8)); dot((-1.5,0,0.8)); dot((-1.2,0,0.8)); dot((-1,0.2,0.2)); dot((-1,0.8,0.8)); dot((-1.2,0.2,1)); dot((-1.5,0.5,1)); dot((-1.8,0.8,1)); [/asy]

Solution

Problem 15

In the diagram, the area of rectangle $PQRS$ is $24$. If $TQ = TR$, what is the area of quadrilateral $PTRS$?

$\text{(A)}\ 18 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 16 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 15$

[asy] size(100); draw((0,0)--(6,0)--(6,4)--(0,4)--cycle); draw((0,4)--(6,2)); draw((5.8,1.1)--(6.2,1.1)); draw((5.8,.9)--(6.2,.9)); draw((5.8,3.1)--(6.2,3.1)); draw((5.8,2.9)--(6.2,2.9)); label("$P$",(0,4),NW); label("$S$",(0,0),SW); label("$R$",(6,0),SE); label("$T$",(6,2),E); label("$Q$",(6,4),NE); [/asy]

Solution

Problem 16

Nicholas is counting the sheep in a flock as they cross a road. The sheep begin to cross the road at 2:00 p.m. and cross at a constant rate of three sheep per minute. After counting $42$ sheep, Nicholas falls asleep. He wakes up an hour and a half later, at which point exactly half of the total flock has crossed the road since 2:00 p.m. How many sheep are there in the entire flock?

$\text{(A)}\ 630 \qquad \text{(B)}\ 621 \qquad \text{(C)}\ 582 \qquad \text{(D)}\ 624 \qquad \text{(E)}\ 618$

Solution

Problem 17

The symbol $\begin{array}{|c|c|}\hline 3 & 4 \\ \hline 5 & 6 \\ \hline \end{array}$ is evaluated as $3 \times 6 + 4 \times 5 = 38$. If $\begin{array}{|c|c|}\hline 2 & 6 \\ \hline 1 &  \\ \hline \end{array}$ is evaluated as $16$, what is the number that should be placed in the empty space?

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Solution

Problem 18

A game is said to be fair if your chance of winning is equal to your chance of losing. How many of the following games, involving tossing a regular six-sided die, are fair?

$\bullet$ You win if you roll a 2

$\bullet$ You win if you roll an even number

$\bullet$ You win if you roll a number less than 4

$\bullet$ You win if you roll a number divisible by 3.

$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$

Solution

Problem 19

Chris and Pat are playing catch. Standing $1 m$ apart, Pat first throws the ball to Chris and then Chris throws the ball back to Pat. Next, standing $2 m$ apart, Pat throws to Chris and Chris throws back to Pat. After each pair of throws, Chris moves $1 m$ farther away from Pat. They stop playing when one of them misses the ball. If the game ends when the $29th$ throw is missed, how far apart are they standing and who misses catching the ball?

$\text{(A)}\ 15 m, Chris \qquad \text{(B)}\ 15 m, Pat \qquad \text{(C)}\ 14 m, Chris \qquad \text{(D)}\ 14 m, Pat \qquad \text{(E)}\ 16 m, Pat$

Solution

Problem 20

While driving at $80 km/h$, Sally’s car passes a hydro pole every four seconds. Which of the following is closest to the distance between two neighbouring hydro poles?

$\text{(A)}\ 50 m \qquad \text{(B)}\ 60 m \qquad \text{(C)}\ 70 m \qquad \text{(D)}\ 80 m \qquad \text{(E)}\ 90 m$

Solution

Part C: Each correct answer is worth 8 points

Problem 21

Emily was at a garage sale where the price of every item was reduced by $10\%$ of its current price every $15$ minutes. At 9:00 a.m., the price of a carpet was $10$ dollars. At 9:15 a.m., the price was reduced to $9$ dollars. As soon as the price of the carpet fell below $8$ dollars, Emily bought it. At what time did Emily buy the carpet?

$\text{(A)}$ 9:45 a.m. $\text{(B)}$ 9:15 a.m. $\text{(C)}$ 9:30 a.m. $\text{(D)}$ 10:15 a.m. $\text{(E)}$ 10:00 a.m.

Solution

Problem 22

In a bin at the Gauss Grocery, the ratio of the number of apples to the number of oranges is $1 : 4$, and the ratio of the number of oranges to the number of lemons is $5 : 2$. What is the ratio of the number of apples to the number of lemons?

$\text{(A)}\ 1 : 2 \qquad \text{(B)}\ 4 : 5 \qquad \text{(C)}\ 5 : 8 \qquad \text{(D)}\ 20 : 8 \qquad \text{(E)}\ 2 : 1$

Solution

Problem 23

Using an equal-armed balance, if $\square\square\square\square$ balances $\bigcirc \bigcirc$ and $\bigcirc \bigcirc \bigcirc$ balances $\triangle \triangle$, which of the following would not balance $\triangle \bigcirc \square$?

$\text{(A)}\ \triangle \bigcirc  \square \qquad \text{(B)}\ \square  \square  \square  \triangle \qquad \text{(C)}\ \square  \square  \bigcirc  \bigcirc \qquad \text{(D)}\ \triangle  \triangle  \square \qquad \text{(E)}\ \bigcirc  \square  \square  \square  \square$

Solution

Problem 24

On a circular track, Alphonse is at point $A$ and Beryl is diametrically opposite at point $B$. Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the same three places on the track. What is the ratio of their speeds?

$\text{(A)}\ 3 : 2 \qquad \text{(B)}\ 3 : 1 \qquad \text{(C)}\ 4 : 1 \qquad \text{(D)}\ 2 : 1 \qquad \text{(E)}\ 5 : 2$

[asy] draw(Circle((0,0),4)); dot((0,4)); dot((0,-4)); label("$A$",(0,4),N); label("$B$",(0,-4),S); draw((-0.5,-4.5)--(-1.5,-4),EndArrow); draw((-0.5,4.5)--(-1.5,4),EndArrow); [/asy]

Solution

Problem 25

How many different combinations of pennies, nickels, dimes and quarters use $48$ coins to total $1$ dollar?

$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Solution

See also

2005 CEMC Gauss (Grade 7) (ProblemsAnswer KeyResources)
Preceded by
2004 CEMC Gauss (Grade 7)
Followed by
2006 CEMC Gauss (Grade 7)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
CEMC Gauss (Grade 7)


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