1988 年 AMC 8 真题（P版）

刻度在 \(10\) 和 \(11\) 之间分成四等份，所以中间刻度分别是 \(10.25\)、\(10.5\) 和 \(10.75\)。

箭头指向刚超过 \(10.25\) 的位置，所以读数在 \(10.25\) 和 \(10.5\) 之间。选项中只有 \(10.3\) 在这个范围内。

所以正确答案是 D。

English

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 .

把小数写成分数，原式变为 \(8 \times \dfrac14 \times 2 \times \dfrac18\)。

分组得 \(\left(8 \times \dfrac18\right)\left(\dfrac14 \times 2\right)\) \(= 1 \times \dfrac12\) \(= \dfrac12\)。

所以正确答案是 C。

English

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 .

每个分数都等于 \(\dfrac{1}{10} = 0.1\)，因为 \(\dfrac{2}{20} = \dfrac{3}{30} = \dfrac{1}{10}\)。

所以和是 \(0.1 + 0.1 + 0.1 = 0.3\)。

所以正确答案是 D。

English

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 .

每一水平行都是深色、浅色、深色这样交替排列，并且以深色开始、以深色结束。所以每一行深色正方形都恰好比浅色正方形多一个。

图形有 \(8\) 行，所以深色正方形比浅色正方形多 \(8\) 个。

所以正确答案是 B。

English

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 .

从量角器读数可见，射线 \(BA\) 约在 \(20^\circ\)，射线 \(BD\) 约在 \(160^\circ\)。

因为 \(\angle CBD\) 是直角，所以射线 \(BC\) 在 \(160^\circ - 90^\circ = 70^\circ\)。

因此 \(\angle ABC = 70^\circ - 20^\circ = 50^\circ\)。

所以正确答案是 C。

English

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 .

分子是 \((0.2)^3 = 0.008\)，分母是 \((0.02)^2 = 0.0004\)。

所以值为 \(\dfrac{0.008}{0.0004} = \dfrac{80}{4} = 20\)。

所以正确答案是 E。

English

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 .

近似每个因数：\(2.46 \approx 2.5\)，\(8.163 \approx 8\)，且 \(5.17 + 4.829 \approx 10\)。

乘积约为 \(2.5 \times 8 \times 10 = 200\)。

所以正确答案是 B。

English

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 .

因数 \(0.075\) 和 \(2.56\) 分别有 \(3\) 位和 \(2\) 位小数，共 \(5\) 位。

从 \(19200\) 右边数 \(5\) 位放小数点，得到 \(0.19200 = 0.192\)。

所以正确答案是 B。

English

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 .

从方格中读出每个三角形的边长。水平或竖直边可直接数格子；一条斜边横跨 \(a\) 个单位、竖跨 \(b\) 个单位，长度为 \(\sqrt{a^2 + b^2}\)。

左上三角形：两条斜边都横跨 \(1\)、竖跨 \(2\)，所以长度都为 \(\sqrt{5}\)，底边长 \(2\)。有两边相等，所以是等腰三角形。

上方中间三角形：它有一条长 \(2\) 的竖边和一条长 \(2\) 的横边（第三边是斜边 \(2\sqrt{2}\)）。有两边相等，所以是等腰三角形。

右上三角形：边长是 \(2\)、\(\sqrt{5}\) 和 \(\sqrt{13}\)，三者都不同，所以不是等腰三角形。

左下（宽而扁的）三角形：两条斜边都横跨 \(3\)、竖跨 \(1\)，所以长度都为 \(\sqrt{10}\)。有两边相等，所以是等腰三角形。

右下三角形：其中两条边都横跨 \(2\) 和 \(1\)（长度为 \(\sqrt{5}\)），第三边为 \(\sqrt{10}\)。有两边相等，所以是等腰三角形。

五个三角形中有四个是等腰三角形。

所以正确答案是 D。

English

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 .

星期每 \(7\) 天重复。因为 \(60 = 8 \times 7 + 4\)，所以 \(60\) 天后是星期四后的第 \(4\) 天。

从星期四往后数：星期五、星期六、星期日、星期一。所以是星期一。

所以正确答案是 A。

English

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 .

因为 \(12^2 = 144\)，\(13^2 = 169\)，且 \(144 \lt 164 \lt 169\)，所以 \(\sqrt{164}\) 位于 \(12\) 和 \(13\) 之间。

所以正确答案是 E。

English

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 .

每个人分摊的费用是 \[\dfrac{20 \times 10^9}{250 \times 10^6} = \dfrac{2 \times 10^{10}}{2.5 \times 10^{8}}\text{。}\]

这等于 \(0.8 \times 10^{2} = 80\)。

所以正确答案是 C。

English

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 .

玫瑰丛围绕圆形边缘，所以数量约等于周长 \(2\pi r = 2\pi(12)\)。

取 \(\pi \approx 3.14\)，约为 \(2 \times 3.14 \times 12 \approx 75\) 英尺，所以大约需要 \(75\) 丛。

所以正确答案是 D。

English

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 .

\(36\) 的因数对是 \(1 \times 36\)、\(2 \times 18\)、\(3 \times 12\)、\(4 \times 9\) 和 \(6 \times 6\)。

最分散的一对给出最大和，\(1 + 36 = 37\)。

所以正确答案是 E。

English

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 .

相加得 \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\)。

\(\dfrac{5}{6}\) 的倒数是 \(\dfrac{6}{5}\)。

所以正确答案是 C。

English

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 .

放 \(6\) 个 X 是可以做到的：把一条主对角线上的三个格子留空，填满其余六格。这样每一行和每一列都少一个格子，使用到的那条对角线有空格，另一条对角线经过空的中心，所以没有任何一条线上出现三个连成一排的 X。

七个 X 不可能：那时只有两个空格，而两个空格最多只能落在三行中的两行，迫使剩下的一行被三个 X 完全填满。

所以最大数是 \(6\)。

所以正确答案是 E。

English

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 .

水平矩形面积为 \(10 \times 2 = 20\)，竖直矩形面积为 \(3 \times 8 = 24\)。它们的重叠部分是一个 \(3 \times 2\) 矩形，面积为 \(6\)。

直接相加两个矩形会把重叠部分算两次，所以阴影面积是 \(20 + 24 - 6 = 38\)。

所以正确答案是 B。

English

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 .

总重量是 \(6 \times 150\) \(+ 4 \times 120\) \(= 900 + 480\) \(= 1380\) 磅。

\(10\) 个孩子的平均体重是 \(\dfrac{1380}{10} = 138\) 磅。

所以正确答案是 C。

English

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 .

这个数列从 \(1\) 开始，公差为 \(4\)。第 \(100\) 项是在第一项基础上加 \(4\) 共 \(99\) 次得到的。

所以第 \(100\) 项是 \(1 + 99 \times 4 = 1 + 396 = 397\)。

所以正确答案是 A。

English

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 .

设满容量为 \(n\)。则 \(\dfrac{45}{n} = \dfrac{36}{100}\)。

解得 \(n = \dfrac{45 \times 100}{36} = 125\) 杯。

所以正确答案是 C。

English

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 .

五个数的和是 \(28 + n\)，平均数是 \(\dfrac{28 + n}{5}\)。中位数是排序后的中间值，依 \(n\) 的大小可能是 \(6\)、\(n\) 或 \(9\)。

如果中位数是 \(6\)：\(\dfrac{28 + n}{5} = 6\)，得 \(n = 2\)，确实小于 \(6\)。如果中位数是 \(n\)：\(\dfrac{28 + n}{5} = n\)，得 \(n = 7\)，确实在 \(6\) 和 \(9\) 之间。如果中位数是 \(9\)：\(\dfrac{28 + n}{5} = 9\)，得 \(n = 17\)，确实大于 \(9\)。

每种情况都给出一个有效值，所以 \(n\) 可以是 \(2\)、\(7\) 或 \(17\)，共有三个可能值。

所以正确答案是 C。

English

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 .

提价 \(25\%\) 会把价格乘以 \(1.25\)，然后打 \(20\%\) 折扣会乘以 \(0.80\)。

总体因数是 \(1.25 \times 0.80 = 1.00\)，所以促销价等于原价。

所以正确答案是 E。

English

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 .

按打计算磁盘，因为 \(12\) 同时是 \(4\) 和 \(3\) 的倍数。一打磁盘买入需 \(3 \times \$5 = \$15\)，卖出可得 \(4 \times \$5 = \$20\)，每打利润为 \(\$5\)。

要获利 \(\$100\)，她需要 \(\dfrac{100}{5} = 20\) 打，也就是 \(20 \times 12 = 240\) 张磁盘。

所以正确答案是 D。

English

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 .

正方形每滚过六边形的一条边，就顺时针转 \(150^\circ\)。（在旋转顶点处，正方形转过 \(360^\circ\) 减去自身 \(90^\circ\) 的角，再减去六边形 \(120^\circ\) 的内角，剩下 \(150^\circ\)。）

从六边形顶部到底部需要滚动 \(3\) 次，总转角为 \(3 \times 150^\circ = 450^\circ\)。这是一整圈（\(360^\circ\)）再加顺时针 \(90^\circ\)。

三角形开始时指向正上方，所以额外顺时针转四分之一圈后，它指向右方。

所以正确答案是 A。

English

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 .

对一位数小时 \(h\)（从 \(1\) 到 \(9\)），显示 \(h\!:\!m_1 m_2\) 倒读相同时需要 \(m_2 = h\)。十位分钟 \(m_1\) 可以是 \(0\)、\(1\)、\(2\)、\(3\)、\(4\) 或 \(5\)，所以每小时有 \(6\) 个回文时间，共 \(9 \times 6 = 54\) 个。

对两位数小时 \(10\)、\(11\)、\(12\)，显示 \(h_1 h_2\!:\!m_1 m_2\) 是回文当且仅当 \(m_1 = h_2\) 且 \(m_2 = h_1\)，正好得到 \(10{:}01\)、\(11{:}11\) 和 \(12{:}21\)，每个小时一个，共再加 \(3\) 个。

总数是 \(54 + 3 = 57\)。

所以正确答案是 A。

English

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 .