1988 AMC 8 — Problems & Online Timed Exam

Timed

40:00

1.

The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of

$10.05$

$10.15$

$10.25$

$10.3$

$10.6$

Answer: D

Concepts:estimation

Difficulty rating: 370

Small Hint:

The marks between $10$ and $11$ divide that unit into four equal parts, so each small division is $0.25$

Big Hint:

The arrow sits just past the first mark after $10,$ so the reading is a little more than $10.25$

Solution:

The scale is divided into fourths between $10$ and $11,$ so the interior tick marks fall at $10.25,$ $10.5,$ and $10.75.$

The arrow points just past the $10.25$ mark, so the reading is between $10.25$ and $10.5.$ Of the choices, only $10.3$ lies in that range.

Thus, the correct answer is D .

2.

What is the product $8 \times 0.25 \times 2 \times 0.125?$

$\dfrac{1}{8}$

$\dfrac{1}{4}$

$\dfrac{1}{2}$

$1$

$2$

Answer: C

Concepts:fraction decimal

Difficulty rating: 450

Small Hint:

Rewrite the decimals as fractions: $0.25 = \dfrac14$ and $0.125 = \dfrac18$

Big Hint:

Pair $8 \times \dfrac18$ and $\dfrac14 \times 2$ to simplify

Solution:

Write the decimals as fractions: $8 \times \dfrac14 \times 2 \times \dfrac18.$

Grouping, $\left(8 \times \dfrac18\right)\left(\dfrac14 \times 2\right)$ $= 1 \times \dfrac12$ $= \dfrac12.$

Thus, the correct answer is C .

3.

The value of

$\dfrac{1}{10} + \dfrac{2}{20} + \dfrac{3}{30}$

is

$0.1$

$0.123$

$0.2$

$0.3$

$0.6$

Answer: D

Concepts:fraction

Difficulty rating: 450

Small Hint:

Each fraction reduces to the same value

Big Hint:

$\dfrac{2}{20} = \dfrac{1}{10}$ and $\dfrac{3}{30} = \dfrac{1}{10}$

Solution:

Each fraction equals $\dfrac{1}{10} = 0.1,$ since $\dfrac{2}{20} = \dfrac{3}{30} = \dfrac{1}{10}.$

So the sum is $0.1 + 0.1 + 0.1 = 0.3.$

Thus, the correct answer is D .

4.

The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by

$7$

$8$

$9$

$10$

$11$

Answer: B

Concepts:counting shapes in figures pattern recognition

Difficulty rating: 560

Small Hint:

Look at the figure one horizontal row at a time

Big Hint:

In each row the squares alternate but both ends are dark, so every row has exactly one more dark square than light square. Count how many rows there are.

Solution:

In each horizontal row the squares alternate dark, light, dark, and so on, beginning and ending with a dark square. So every row has exactly one more dark square than light square.

The figure has $8$ rows, so the dark squares exceed the light squares by $8.$

Thus, the correct answer is B .

5.

If $\angle CBD$ is a right angle, then this protractor indicates that the measure of $\angle ABC$ is approximately

$20^\circ$

$40^\circ$

$50^\circ$

$70^\circ$

$120^\circ$

Answer: C

Concepts:angle chasing

Difficulty rating: 800

Small Hint:

On the protractor, ray $BA$ reads about $20^\circ$ and ray $BD$ reads about $160^\circ$

Big Hint:

Since $\angle CBD = 90^\circ$ and $BD$ reads $160^\circ,$ ray $BC$ reads $160^\circ - 90^\circ.$ Then $\angle ABC$ is the difference between the readings of $BC$ and $BA.$

Solution:

Reading the protractor, ray $BA$ is at about $20^\circ$ and ray $BD$ is at about $160^\circ.$

Because $\angle CBD$ is a right angle, ray $BC$ is at $160^\circ - 90^\circ = 70^\circ.$

Therefore $\angle ABC = 70^\circ - 20^\circ = 50^\circ.$

Thus, the correct answer is C .

6.

The value of

$\dfrac{(0.2)^3}{(0.02)^2}$

is

$0.2$

$2$

$10$

$15$

$20$

Answer: E

Concepts:decimal exponent

Difficulty rating: 730

Small Hint:

Compute the numerator and denominator separately: $(0.2)^3 = 0.008$

Big Hint:

$(0.02)^2 = 0.0004,$ then divide

Solution:

The numerator is $(0.2)^3 = 0.008$ and the denominator is $(0.02)^2 = 0.0004.$

So the value is $\dfrac{0.008}{0.0004} = \dfrac{80}{4} = 20.$

Thus, the correct answer is E .

7.

The expression $2.46 \times 8.163 \times (5.17 + 4.829)$ is closest to

$100$

$200$

$300$

$400$

$500$

Answer: B

Concepts:estimation

Difficulty rating: 730

Small Hint:

Round each factor to a convenient number: $2.46 \approx 2.5,$ $8.163 \approx 8$

Big Hint:

The sum $5.17 + 4.829 \approx 10,$ so the product is about $2.5 \times 8 \times 10$

Solution:

Round each factor: $2.46 \approx 2.5,$ $8.163 \approx 8,$ and $5.17 + 4.829 \approx 10.$

The product is about $2.5 \times 8 \times 10 = 200.$

Thus, the correct answer is B .

8.

Betty used a calculator to find the product $0.075 \times 2.56.$ She forgot to enter the decimal points. The calculator showed $19200.$ If Betty had entered the decimal points correctly, the answer would have been

$0.0192$

$0.192$

$1.92$

$19.2$

$192$

Answer: B

Concepts:decimal place value

Difficulty rating: 800

Small Hint:

Count the total number of decimal places in $0.075$ and $2.56$

Big Hint:

There are $3 + 2 = 5$ decimal places, so place the decimal point that many digits from the right of $19200$

Solution:

The factors $0.075$ and $2.56$ have $3$ and $2$ decimal places, for a total of $5.$

Placing the decimal point $5$ digits from the right of $19200$ gives $0.19200 = 0.192.$

Thus, the correct answer is B .

9.

An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?

$1$

$2$

$3$

$4$

$5$

Answer: D

Concepts:isosceles triangle distance formula

Difficulty rating: 960

Small Hint:

Find each triangle’s side lengths from the grid: a slanted side spanning $a$ across and $b$ up has length $\sqrt{a^2 + b^2}.$ A triangle is isosceles when two of its sides are equal.

Big Hint:

Work out all three side lengths for each of the five triangles and look for a matching pair; only one triangle ends up with three different side lengths.

Solution:

Read each triangle’s side lengths from the grid. A horizontal or vertical side is counted directly, and a slanted side spanning $a$ units across and $b$ units up has length $\sqrt{a^2 + b^2}.$

Top-left triangle: its two slanted sides each span $1$ across and $2$ up, so each has length $\sqrt{5},$ with a base of length $2.$ Two equal sides, so it is isosceles.

Top-middle triangle: it has a vertical side of length $2$ and a horizontal side of length $2$ (its third side is the slant $2\sqrt{2}$ ). Two equal sides, so it is isosceles.

Top-right triangle: its sides are $2,$ $\sqrt{5},$ and $\sqrt{13},$ which are all different, so it is not isosceles.

Bottom-left (wide, flat) triangle: its two slanted sides each span $3$ across and $1$ up, so each has length $\sqrt{10}.$ Two equal sides, so it is isosceles.

Bottom-right triangle: two of its sides each span $2$ and $1$ (giving length $\sqrt{5}$ ), while the third is $\sqrt{10}.$ Two equal sides, so it is isosceles.

Four of the five triangles are isosceles.

Thus, the correct answer is D .

10.

Chris’ birthday is on a Thursday this year. What day of the week will it be $60$ days after her birthday?

Monday

Wednesday

Thursday

Friday

Saturday

Answer: A

Concepts:modular arithmetic date and time

Difficulty rating: 820

Small Hint:

The day of the week repeats every $7$ days, so only the remainder of $60$ upon division by $7$ matters

Big Hint:

$60 = 8 \times 7 + 4,$ so the day is $4$ days after Thursday

Solution:

Days of the week repeat every $7$ days. Since $60 = 8 \times 7 + 4,$ the day $60$ days later is $4$ days after Thursday.

Counting forward from Thursday: Friday, Saturday, Sunday, Monday. So it is a Monday.

Thus, the correct answer is A .

11.

The value of $\sqrt{164}$ is

$42$

less than $10$

between $10$ and $11$

between $11$ and $12$

between $12$ and $13$

Answer: E

Concepts:perfect square estimation

Difficulty rating: 660

Small Hint:

Find perfect squares just below and just above $164$

Big Hint:

$12^2 = 144$ and $13^2 = 169$

Solution:

Since $12^2 = 144$ and $13^2 = 169,$ and $144 \lt 164 \lt 169,$ the value $\sqrt{164}$ lies between $12$ and $13.$

Thus, the correct answer is E .

12.

Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person’s share is

$\$40$

$\$50$

$\$80$

$\$100$

$\$125$

Answer: C

Concepts:place value

Difficulty rating: 730

Small Hint:

Write the numbers with powers of ten: $20$ billion $= 2 \times 10^{10}$ and $250$ million $= 2.5 \times 10^{8}$

Big Hint:

Divide the total cost by the number of people

Solution:

Each person’s share is $\dfrac{20 \times 10^9}{250 \times 10^6} = \dfrac{2 \times 10^{10}}{2.5 \times 10^{8}}.$

This equals $0.8 \times 10^{2} = 80.$

Thus, the correct answer is C .

13.

If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet?

$12$

$38$

$48$

$75$

$450$

Answer: D

Concepts:circumference estimation

Difficulty rating: 860

Small Hint:

The bushes go around the edge, so find the circumference $2\pi r$

Big Hint:

With $\pi \approx 3.14,$ compute $2 \times 3.14 \times 12$

Solution:

The bushes surround the circular edge, so their number is about the circumference $2\pi r = 2\pi(12).$

Using $\pi \approx 3.14,$ this is about $2 \times 3.14 \times 12 \approx 75$ feet, so roughly $75$ bushes are needed.

Thus, the correct answer is D .

14.

If $\diamond$ and $\triangle$ are whole numbers and $\diamond \times \triangle = 36,$ the largest possible value of $\diamond + \triangle$ is

$12$

$13$

$15$

$20$

$37$

Answer: E

Concepts:factor optimization

Difficulty rating: 730

Small Hint:

List the factor pairs of $36$

Big Hint:

The sum is largest when the two factors are as far apart as possible

Solution:

The factor pairs of $36$ are $1 \times 36,$ $2 \times 18,$ $3 \times 12,$ $4 \times 9,$ and $6 \times 6.$

The sum is largest for the most spread-out pair, $1 + 36 = 37.$

Thus, the correct answer is E .

15.

What is the reciprocal of $\left(\dfrac{1}{2} + \dfrac{1}{3}\right)?$

$\dfrac{1}{6}$

$\dfrac{2}{5}$

$\dfrac{6}{5}$

$\dfrac{5}{2}$

$5$

Answer: C

Concepts:fraction

Difficulty rating: 660

Small Hint:

First add the fractions using a common denominator of $6$

Big Hint:

The reciprocal flips the numerator and denominator

Solution:

Adding, $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}.$

The reciprocal of $\dfrac{5}{6}$ is $\dfrac{6}{5}.$

Thus, the correct answer is C .

16.

Placing no more than one X in each small square, what is the greatest number of X’s that can be put on a $3 \times 3$ grid without getting three X’s in a row vertically, horizontally, or diagonally?

$2$

$3$

$4$

$5$

$6$

Answer: E

Concepts:extremal argument pigeonhole principle

Difficulty rating: 1120

Small Hint:

Try to leave one square empty in each row and column so no line gets filled

Big Hint:

With $7$ X’s only two squares are empty, but two empty squares cannot break all three rows, so some row would be full

Solution:

A placement of $6$ X’s works: leave the three squares along one main diagonal empty and fill the other six. Then each row and each column is missing one square, the used diagonal has an empty square, and the other diagonal passes through the empty center, so no line of three is complete.

Seven X’s is impossible: only two squares would be empty, and two empty squares can lie in at most two of the three rows, forcing the remaining row to be completely filled with three X’s.

So the greatest number is $6.$

Thus, the correct answer is E .

17.

The shaded area is formed by two intersecting perpendicular rectangles, with dimensions as shown. Its area, in square units, is

$23$

$38$

$44$

$46$

unable to be determined from the information given

Answer: B

Concepts:area inclusion-exclusion

Difficulty rating: 960

Small Hint:

Add the areas of the two rectangles, but the overlapping region gets counted twice

Big Hint:

The horizontal rectangle is $10 \times 2,$ the vertical one is $3 \times 8,$ and the overlap is $3 \times 2$

Solution:

The horizontal rectangle has area $10 \times 2 = 20,$ and the vertical rectangle has area $3 \times 8 = 24.$ Their overlap, a $3 \times 2$ rectangle, has area $6.$

Adding the two rectangles counts the overlap twice, so the shaded area is $20 + 24 - 6 = 38.$

Thus, the correct answer is B .

18.

The average weight of $6$ boys is $150$ pounds and the average weight of $4$ girls is $120$ pounds. The average weight of the $10$ children is

$135$ pounds

$137$ pounds

$138$ pounds

$140$ pounds

$141$ pounds

Answer: C

Concepts:mean

Difficulty rating: 860

Small Hint:

Find the total weight of the boys and the total weight of the girls separately

Big Hint:

Divide the combined total weight by $10$

Solution:

The total weight is $6 \times 150$ $+ 4 \times 120$ $= 900 + 480$ $= 1380$ pounds.

The average of the $10$ children is $\dfrac{1380}{10} = 138$ pounds.

Thus, the correct answer is C .

19.

What is the $100$ th number in the arithmetic sequence $1,$ $5,$ $9,$ $13,$ $17,$ $21,$ $25,$ $\ldots?$

$397$

$399$

$401$

$403$

$405$

Answer: A

Concepts:arithmetic sequence

Difficulty rating: 820

Small Hint:

The common difference is $4,$ so each term adds $4$ to the previous one

Big Hint:

Starting from $1,$ reaching the $100$ th term means adding $4$ a total of $99$ times

Solution:

The sequence starts at $1$ with common difference $4.$ The $100$ th term is reached by adding $4$ to the first term $99$ times.

So the $100$ th term is $1 + 99 \times 4 = 1 + 396 = 397.$

Thus, the correct answer is A .

20.

The glass gauge on a cylindrical coffee maker shows there are $45$ cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full?

$80$

$100$

$125$

$130$

$262$

Answer: C

Concepts:percentage ratio and proportion

Difficulty rating: 930

Small Hint:

Set up a proportion: $45$ cups is to the full amount as $36$ is to $100$

Big Hint:

If $36\%$ is $45$ cups, first find how many cups $4\%$ is

Solution:

Let $n$ be the full capacity. Then $\dfrac{45}{n} = \dfrac{36}{100}.$

Solving, $n = \dfrac{45 \times 100}{36} = 125$ cups.

Thus, the correct answer is C .

21.

A fifth number, $n,$ is added to the set of numbers $\{3, 6, 9, 10\}$ to make the mean of the set of five numbers equal to its median. The number of possible values for $n$ is

$1$

$2$

$3$

$4$

more than $4$

Answer: C

Concepts:mean median (data)casework

Difficulty rating: 1260

Small Hint:

The mean of the five numbers is $\dfrac{28 + n}{5};$ the median depends on where $n$ lands in the sorted order

Big Hint:

Split into cases: the median is $6,$ is $n$ itself, or is $9,$ and solve the equation mean $=$ median in each

Solution:

The five numbers have sum $28 + n$ and mean $\dfrac{28 + n}{5}.$ The median is the middle value when sorted, which is $6,$ $n,$ or $9$ depending on the size of $n.$

If the median is $6$ : $\dfrac{28 + n}{5} = 6$ gives $n = 2,$ which is indeed less than $6.$ If the median is $n$ : $\dfrac{28 + n}{5} = n$ gives $n = 7,$ which is between $6$ and $9.$ If the median is $9$ : $\dfrac{28 + n}{5} = 9$ gives $n = 17,$ which is indeed greater than $9.$

Each case yields a valid value, so $n$ can be $2,$ $7,$ or $17$ — three possible values.

Thus, the correct answer is C .

22.

Tom’s Hat Shoppe increased all original prices by $25\%.$ Now the shoppe is having a sale where all prices are $20\%$ off these increased prices. Which statement best describes the sale price of an item?

The sale price is $5\%$ higher than the original price.

The sale price is higher than the original price, but by less than $5\%.$

The sale price is higher than the original price, but by more than $5\%.$

The sale price is lower than the original price.

The sale price is the same as the original price.

Answer: E

Concepts:percentage

Difficulty rating: 1060

Small Hint:

Increasing by $25\%$ multiplies the price by $1.25;$ taking $20\%$ off multiplies by $0.80$

Big Hint:

Multiply the two factors together and compare the result to $1$

Solution:

Increasing by $25\%$ multiplies the price by $1.25,$ and then taking $20\%$ off multiplies by $0.80.$

The overall factor is $1.25 \times 0.80 = 1.00,$ so the sale price equals the original price.

Thus, the correct answer is E .

23.

Maria buys computer disks at a price of $4$ for $\$5$ and sells them at a price of $3$ for $\$5.$ How many computer disks must she sell in order to make a profit of $\$100?$

$100$

$120$

$200$

$240$

$1200$

Answer: D

Concepts:ratio and proportion

Difficulty rating: 1110

Small Hint:

Work with a convenient batch of $12$ disks, since $12$ is a multiple of both $4$ and $3$

Big Hint:

A dozen disks costs $\$15$ and sells for $\$20,$ a profit of $\$5$ per dozen

Solution:

Consider disks in dozens, since $12$ is a multiple of both $4$ and $3.$ A dozen disks costs $3 \times \$5 = \$15$ to buy and sells for $4 \times \$5 = \$20,$ for a profit of $\$5$ per dozen.

To make $\$100$ profit she needs $\dfrac{100}{5} = 20$ dozen, which is $20 \times 12 = 240$ disks.

Thus, the correct answer is D .

24.

The square in the first diagram “rolls” clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4?$

Answer: A

Concepts:transformation regular polygon

Difficulty rating: 1510

Small Hint:

Each time the square rolls over one edge of the hexagon it pivots about a shared corner and turns $150^\circ$ clockwise

Big Hint:

Going from the top edge to the bottom edge is $3$ rolls, a total of $3 \times 150^\circ = 450^\circ,$ which is one full turn plus an extra quarter turn clockwise

Solution:

Each roll of the square over one edge of the hexagon turns it $150^\circ$ clockwise. (At the pivot corner the square turns through $360^\circ$ minus the square’s own $90^\circ$ corner minus the hexagon’s $120^\circ$ interior angle, which leaves $150^\circ.$ )

From the top of the hexagon to the bottom is $3$ rolls, for a total of $3 \times 150^\circ = 450^\circ.$ That is one complete revolution ( $360^\circ$ ) plus an extra $90^\circ$ clockwise.

The triangle starts pointing straight up, so after the extra quarter turn clockwise it points to the right.

Thus, the correct answer is A .

25.

A palindrome is a whole number that reads the same forwards as backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are $1{:}01,$ $4{:}44,$ and $12{:}21.$ How many times during a $12$ -hour period will be palindromes?

$57$

$60$

$63$

$90$

$93$

Answer: A

Concepts:palindrome casework

Difficulty rating: 1380

Small Hint:

Split into single-digit hours ( $1$ through $9$ ) and two-digit hours ( $10,$ $11,$ $12$ ), since the number of digits differs

Big Hint:

For a single-digit hour $h,$ the time $h\!:\!m_1 m_2$ is a palindrome when $m_2 = h,$ leaving $m_1$ free among $0$ through $5$

Solution:

For a single-digit hour $h$ (from $1$ to $9$ ), the display $h\!:\!m_1 m_2$ reads the same backwards when $m_2 = h.$ The tens digit $m_1$ can be $0,$ $1,$ $2,$ $3,$ $4,$ or $5,$ giving $6$ palindromes per hour, so $9 \times 6 = 54.$

For the two-digit hours $10,$ $11,$ $12,$ the display $h_1 h_2\!:\!m_1 m_2$ is a palindrome when $m_1 = h_2$ and $m_2 = h_1,$ giving exactly $10{:}01,$ $11{:}11,$ and $12{:}21$ — one each, for $3$ more.

The total is $54 + 3 = 57.$

Thus, the correct answer is A .