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

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

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

Solution

Problem 19

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

Solution

Problem 20

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

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

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

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1998 CEMC Gauss (Grade 7) Problems

Contents

Part A: Each correct answer is worth 5 points

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Part B: Each correct answer is worth 6 points

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Part C: Each correct answer is worth 8 points

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also