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Difference between revisions of "1998 CEMC Gauss (Grade 7) Problems"

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= Part A: Each correct answer is worth 5 points =
 
= Part A: Each correct answer is worth 5 points =
 +
 +
*If any diagrams are missing, consult the pdf: https://www.cemc.uwaterloo.ca/contests/past_contests/1998/1998Gauss7Contest.pdf
  
 
== Problem 1 ==
 
== Problem 1 ==
Line 71: Line 73:
  
 
== Problem 9 ==
 
== Problem 9 ==
 +
Two numbers have a sum of 32. If one of the numbers is – 36, what is the other number?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 68 \qquad \text{(B)}\ -4 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 72 \qquad \text{(E)}\ -68 </math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 9|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
 +
At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by 0.25 seconds. If Bonnie’s time was exactly 7.80 seconds, how long did it take for Wendy to go down the slide?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 7.8 \qquad \text{(B)}\ 8.05 \qquad \text{(C)}\ 7.55 \qquad \text{(D)}\ 7.15 \qquad \text{(E)}\ 7.5</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 10|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 10|Solution]]
Line 85: Line 89:
  
 
== Problem 11 ==
 
== Problem 11 ==
 +
Kalyn cut rectangle R from a sheet of paper. A smaller rectangle is then cut from the large rectangle R to produce figure S. In comparing R to S,
 +
 +
[R is a rectangle with sides 8 and 6 cm. S is the same as R with a 4x1 rectangle cut from one of its corners.]
 +
 +
<math>\text{(A) the area and perimeter both decrease}</math>
 +
 +
<math>\text{(B) the area decreases and the perimeter increases}</math>
 +
 +
<math>\text{(C) the area and perimeter both increase}</math>
 +
 +
<math>\text{(D) the area increases and the perimeter decreases}</math>
  
−
<math>\text{(A)}\  \qquad \text{(B)}\  \qquad \text{(C)}\  \qquad \text{(D)}\  \qquad \text{(E)}\ </math>
+
<math>\text{(E) the area decreases and the perimeter stays the same}</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 11|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 +
Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 1 \dfrac{1}{4} \text{h} \qquad \text{(B)}\ 3 \text{h} \qquad \text{(C)}\ 5 \text{h} \qquad \text{(D)}\ 10 \text{h} \qquad \text{(E)}\ 12 \dfrac{1}{2} \text{h}</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 12|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 +
The pattern of figures (triangle, dark circle, square, dark triangle, circle) is repeated over and over again. The 214th figure in the sequence is
 +
 +
<math>\text{(A) triangle}</math>
 +
 +
<math>\text{(B) dark circle}</math>
 +
 +
<math>\text{(C) square}</math>
 +
 +
<math>\text{(D) dark triangle}</math>
  
−
<math>\text{(A)}\  \qquad \text{(B)}\  \qquad \text{(C)}\  \qquad \text{(D)}\  \qquad \text{(E)}\ </math>
+
<math>\text{(E) circle}</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 13|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 +
A cube has a volume of <math>125 \text{cm}^3.</math> What is the area of one face of the cube?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
 
 +
<math>\text{(A)}\ 20 \text{cm}^2 \qquad \text{(B)}\ 25 \text{cm}^2 \qquad \text{(C)}\ 41 \dfrac{2}{3} \text{cm}^2 \qquad \text{(D)}\ 5 \text{cm}^2 \qquad \text{(E)}\ 75 \text{cm}^2 </math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 14|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 +
The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of n?
 +
 +
[A 3x3 magic square grid is shown. 8 is in the 1st row 1st column. 9 is in the 2nd row 1st column. 4 is in the 2nd row 3rd column. 4 is in the 3rd row 1st column. <math>n</math> is in the 3rd row 2nd column.]
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 15|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 15|Solution]]
  
 
== Problem 16 ==
 
== Problem 16 ==
 +
Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two digit number is subtracted from the three digit number, what is the smallest difference?
 +
 +
[Align three boxes to the right and two boxes below so it looks like a three digit number subtracting a two digit number.]
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 269 \qquad \text{(B)}\ 278 \qquad \text{(C)}\ 484 \qquad \text{(D)}\ 271 \qquad \text{(E)}\ 261</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 16|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 16|Solution]]
  
 
== Problem 17 ==
 
== Problem 17 ==
 +
Claire takes a square piece of paper and folds it in half four times without unfolding, making an
 +
isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the
 +
paper would look like:
 +
 +
(see attached pdf for diagrams)
  
 
<math>\text{(A)}\  \qquad \text{(B)}\  \qquad \text{(C)}\  \qquad \text{(D)}\  \qquad \text{(E)}\ </math>
 
<math>\text{(A)}\  \qquad \text{(B)}\  \qquad \text{(C)}\  \qquad \text{(D)}\  \qquad \text{(E)}\ </math>
Line 127: Line 165:
  
 
== Problem 18 ==
 
== Problem 18 ==
 +
The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown:
 +
<cmath>1. \quad AUSSG \qquad 9981</cmath>
 +
<cmath>2. \quad USSGA \qquad 9819</cmath>
 +
<cmath>3. \quad SSGAU \qquad 8199</cmath>
 +
If the pattern continues in this way, what number will appear in front of GAUSS 1998?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 18|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 18|Solution]]
  
 
== Problem 19 ==
 
== Problem 19 ==
 +
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 7 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 11</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 19|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 19|Solution]]
  
 
== Problem 20 ==
 
== Problem 20 ==
 +
Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one
 +
red edge. What is the smallest number of red edges?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
 
 +
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 20|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 20|Solution]]
Line 148: Line 195:
 
== Problem 21 ==
 
== Problem 21 ==
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
Ten points are spaced equally around a circle. How many different chords can be formed by joining
 +
any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)
 +
 
 +
<math>\text{(A)}\ 9 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 66 \qquad \text{(E)}\ 55 </math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 21|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 21|Solution]]
Line 154: Line 204:
 
== Problem 22 ==
 
== Problem 22 ==
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
Each time a bar of soap is used, its volume decreases by 10%. What is the minimum number of times
 +
a new bar would have to be used so that less than one-half its volume remains?
 +
 
 +
<math>\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 22|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 22|Solution]]
  
 
== Problem 23 ==
 
== Problem 23 ==
 +
A cube measures 10 by 10 by 10 cm. Three cuts are made parallel to the faces of the cube as shown (in three perpendicular directions) creating eight separate solids which are then separated. What is the increase in the total surface area?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 300 \text{cm}^2 \qquad \text{(B)}\ 800 \text{cm}^2 \qquad \text{(C)}\ 1200 \text{cm}^2 \qquad \text{(D)}\ 600 \text{cm}^2 \qquad \text{(E)}\ 0 \text{cm}^2</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 23|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 23|Solution]]
  
 
== Problem 24 ==
 
== Problem 24 ==
 +
On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws?
  
−
<math>\text{(A)}\ \qquad \text{(B)}\ \qquad \text{(C)}\ \qquad \text{(D)}\ \qquad \text{(E)}\ </math>
+
<math>\text{(A)}\ 38 \qquad \text{(B)}\ 39 \qquad \text{(C)}\ 54 \qquad \text{(D)}\ 55 \qquad \text{(E)}\ 50</math>
  
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 24|Solution]]
 
[[1998 CEMC Gauss (Grade 7) Problems/Problem 24|Solution]]

Latest revision as of 15:43, 29 January 2021

Part A: Each correct answer is worth 5 points

Problem 1

The value of $\frac{1998 - 998}{1000}$ is

$\text{(A)}\ 1 \qquad \text{(B)}\ 1000 \qquad \text{(C)}\ 0.1 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 0.001$

Solution

Problem 2

The number $4567$ is tripled. The ones digit (units digit) in the resulting number is

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 1$

Solution

Problem 3

If $S = 6\times 10,000 + 5\times 1,000 + 4\times 10 + 3\times 1$, what is $S$?

$\text{(A)}\ 6,543 \qquad \text{(B)}\ 65,043 \qquad \text{(C)}\ 65,431 \qquad \text{(D)}\ 65,403 \qquad \text{(E)}\ 60,541$

Solution

Problem 4

Jean writes five tests and achieves the marks shown on the graph. What is her average mark on these five tests?

[insert bar graph with 5 bars: 80, 70, 60, 90, 80]

$\text{(A)}\ 74 \qquad \text{(B)}\ 76 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 79$

Solution

Problem 5

If a machine produces 150 items in one minute, how many would it produce in 10 seconds?

$\text{(A)}\ 10 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 25 \qquad \text{(E)}\ 30$

Solution

Problem 6

In the multiplication question, the sum of the digits in the four boxes is:

[Multiply $879 \times 492$ using long multiplication. Find the sum of the four numbers in the thousands place column.]

$\text{(A)}\ 13 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 27 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 22$

Solution

Problem 7

A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m apart along the 4 sides of the field, how many posts are needed to completely fence the field?

$\text{(A)}\ 24 \qquad \text{(B)}\ 26 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 32$

Solution

Problem 8

Tuesday’s high temperature was 4 C warmer than that of Monday’s. Wednesday’s high temperature was 6 C cooler than that of Monday’s. If Tuesday’s high temperature was 22 C, what was Wednesday’s high temperature?

$\text{(A)} 20 \quad\text{(B)} 24\quad \text{(C)} 12\quad \text{(D)} 32 \quad \text{(E)} 16 \quad$


(all in Celsius)

Solution

Problem 9

Two numbers have a sum of 32. If one of the numbers is – 36, what is the other number?

$\text{(A)}\ 68 \qquad \text{(B)}\ -4 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 72 \qquad \text{(E)}\ -68$

Solution

Problem 10

At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by 0.25 seconds. If Bonnie’s time was exactly 7.80 seconds, how long did it take for Wendy to go down the slide?

$\text{(A)}\ 7.8 \qquad \text{(B)}\ 8.05 \qquad \text{(C)}\ 7.55 \qquad \text{(D)}\ 7.15 \qquad \text{(E)}\ 7.5$

Solution

Part B: Each correct answer is worth 6 points

Problem 11

Kalyn cut rectangle R from a sheet of paper. A smaller rectangle is then cut from the large rectangle R to produce figure S. In comparing R to S,

[R is a rectangle with sides 8 and 6 cm. S is the same as R with a 4x1 rectangle cut from one of its corners.]

$\text{(A) the area and perimeter both decrease}$

$\text{(B) the area decreases and the perimeter increases}$

$\text{(C) the area and perimeter both increase}$

$\text{(D) the area increases and the perimeter decreases}$

$\text{(E) the area decreases and the perimeter stays the same}$

Solution

Problem 12

Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees?

$\text{(A)}\ 1 \dfrac{1}{4} \text{h} \qquad \text{(B)}\ 3 \text{h} \qquad \text{(C)}\ 5 \text{h} \qquad \text{(D)}\ 10 \text{h} \qquad \text{(E)}\ 12 \dfrac{1}{2} \text{h}$

Solution

Problem 13

The pattern of figures (triangle, dark circle, square, dark triangle, circle) is repeated over and over again. The 214th figure in the sequence is

$\text{(A) triangle}$

$\text{(B) dark circle}$

$\text{(C) square}$

$\text{(D) dark triangle}$

$\text{(E) circle}$

Solution

Problem 14

A cube has a volume of $125 \text{cm}^3.$ What is the area of one face of the cube?


$\text{(A)}\ 20 \text{cm}^2 \qquad \text{(B)}\ 25 \text{cm}^2 \qquad \text{(C)}\ 41 \dfrac{2}{3} \text{cm}^2 \qquad \text{(D)}\ 5 \text{cm}^2 \qquad \text{(E)}\ 75 \text{cm}^2$

Solution

Problem 15

The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of n?

[A 3x3 magic square grid is shown. 8 is in the 1st row 1st column. 9 is in the 2nd row 1st column. 4 is in the 2nd row 3rd column. 4 is in the 3rd row 1st column. $n$ is in the 3rd row 2nd column.]

$\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$

Solution

Problem 16

Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two digit number is subtracted from the three digit number, what is the smallest difference?

[Align three boxes to the right and two boxes below so it looks like a three digit number subtracting a two digit number.]

$\text{(A)}\ 269 \qquad \text{(B)}\ 278 \qquad \text{(C)}\ 484 \qquad \text{(D)}\ 271 \qquad \text{(E)}\ 261$

Solution

Problem 17

Claire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the paper would look like:

(see attached pdf for diagrams)

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

Solution

Problem 18

The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown: \[1. \quad AUSSG \qquad 9981\] \[2. \quad USSGA \qquad 9819\] \[3. \quad SSGAU \qquad 8199\] If the pattern continues in this way, what number will appear in front of GAUSS 1998?

$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

Solution

Problem 19

Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?

$\text{(A)}\ 7 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 11$

Solution

Problem 20

Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one red edge. What is the smallest number of red edges?


$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

Solution

Part C: Each correct answer is worth 8 points

Problem 21

Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.)

$\text{(A)}\ 9 \qquad \text{(B)}\ 45 \qquad \text{(C)}\ 17 \qquad \text{(D)}\ 66 \qquad \text{(E)}\ 55$

Solution

Problem 22

Each time a bar of soap is used, its volume decreases by 10%. What is the minimum number of times a new bar would have to be used so that less than one-half its volume remains?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Solution

Problem 23

A cube measures 10 by 10 by 10 cm. Three cuts are made parallel to the faces of the cube as shown (in three perpendicular directions) creating eight separate solids which are then separated. What is the increase in the total surface area?

$\text{(A)}\ 300 \text{cm}^2 \qquad \text{(B)}\ 800 \text{cm}^2 \qquad \text{(C)}\ 1200 \text{cm}^2 \qquad \text{(D)}\ 600 \text{cm}^2 \qquad \text{(E)}\ 0 \text{cm}^2$

Solution

Problem 24

On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws?

$\text{(A)}\ 38 \qquad \text{(B)}\ 39 \qquad \text{(C)}\ 54 \qquad \text{(D)}\ 55 \qquad \text{(E)}\ 50$

Solution

Problem 25

Two natural numbers, $p$ and $q,$ do not end in zero. The product of any pair, $p$ and $q,$ is a power of 10 (that is, 10, 100, 1000, 10 000 , ...). If $p > q$ , the last digit of $p-q$ cannot be

$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

Solution

See also

1998 CEMC Gauss (Grade 7) (Problems • Answer Key • Resources)
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CEMC Gauss (Grade 7)
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