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AQA · GCSE · Mathematics 주제 1 8:31 영어 내레이션 · 영어 + 중국어 자막 burned-in
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Topic 1 is the number toolkit for everything else.
第 1 主题是其他一切的数字工具箱。
Prime factors give the HCF from the lowest shared powers and the LCM from the highest.
质因数分解后，最大公因数取共有质因数的最低次幂，最小公倍数取最高次幂。
The index laws, with negative and fractional powers at Higher.
指数法则，高级卷还有负指数和分数指数。
A rounded value lies half a unit either side.
近似值在上下各半个单位内。
And a percentage change is a multiplier.
百分比变化就是乘数。
Read a balance on a number line.
在数轴上看余额。
Start at minus 12.
从 −12 开始。
A credit of 35 moves right to 23; a debit of 9 moves left, so the balance is 14 yuan.
收入 35 向右移到 23；支出 9 向左移，余额是 14 元。
Subtracting a negative moves right: minus 4 minus minus 7 is 3.
减去负数向右移：−4 − (−7) = 3。
And watch the square: minus 3 squared is minus 9, but minus 3 in brackets, squared, is 9.
注意平方：−3² = −9，但 (−3)² = 9。
Write each number as primes: 84 is 2 squared times 3 times 7, and 120 is 2 cubed times 3 times 5.
把每个数写成质因数：84 = 2² × 3 × 7，120 = 2³ × 3 × 5。
The shared primes 2, 2 and 3 go in the overlap: their product, 12, is the HCF.
共有的质因数 2、2、3 放在重叠区：乘积 12 就是最大公因数。
84 keeps its 7 and 120 keeps 2 and 5; everything multiplied is the LCM, 840.
84 剩下 7，120 剩下 2 和 5；全部相乘得最小公倍数 840。
Check: 12 times 840 equals 84 times 120.
检验：12 × 840 = 84 × 120。
Choose a drink and a fruit.
选一种饮料和一种水果。
Fix tea first and run through apple, banana and melon; then juice.
先固定茶，依次配苹果、香蕉、甜瓜；再换果汁。
The grid shows every outcome once: 2 times 3 is 6.
表格把每种结果各列一次：2 × 3 = 6。
If tea-melon is unavailable, cross it out: 5 remain.
如果没有茶配甜瓜，就划掉：剩 5 种。
The product rule is the Higher shortcut, but a restriction can break it, so check.
乘法原理是高级卷的捷径，但限制条件可能使它失效，所以要检查。
Read a fractional index from the bottom up.
分数指数从下往上读。
The denominator is a root and the numerator is a power: the fourth root of 16 is 2, and 2 cubed is 8.
分母表示开方，分子表示乘方：16 的四次方根是 2，2³ = 8。
27 to the two thirds is 3 squared, 9.
27^{2/3} = 3² = 9。
A minus sign means reciprocal: 16 to the minus a half is one quarter.
负号表示倒数：16^{−1/2} = ¼。
Higher only.
仅限高级卷。
To multiply, multiply the numbers and add the powers: 12 times 10 squared, then renormalise to 1.2 times 10 cubed.
相乘时，数字相乘、指数相加：12 × 10²，再规范为 1.2 × 10³。
To divide, divide and subtract: 3 times 10 cubed.
相除时，数字相除、指数相减：3 × 10³。
To add, match the powers first: 3.2 times 10 to the 4.
相加时先统一指数：3.2 × 10⁴。
A square of area 12 has side root 12.
面积为 12 的正方形边长是 √12。
Since 12 is 4 times 3, the side is root 4 times root 3, which is 2 root 3 centimetres, exactly.
因为 12 = 4 × 3，边长是 √4 × √3 = 2√3 厘米，精确值。
Root 27 is 3 root 3, so the sum is 5 root 3.
√27 = 3√3，所以和是 5√3。
To rationalise 1 over 3 plus root 2, multiply by 3 minus root 2 over itself: 3 minus root 2, over 7.
要使 1/(3 + √2) 分母有理化，乘以 (3 − √2)/(3 − √2)：得 (3 − √2)/7。
Rounded to 8.0 at one decimal place, a length lies half a unit either side: from 7.95, included, up to 8.05, not included.
一位小数四舍五入为 8.0 的长度，在上下各半个单位内：从 7.95（含）到 8.05（不含）。
If it was truncated instead, it lies from 8.0 up to 8.1.
如果是截断得到的，则在 8.0 到 8.1 之间。
Same reading, different intervals.
读数相同，区间不同。
And 3.6 kilograms to 1 decimal place means 3.55 up to 3.65.
3.6 千克（一位小数）表示 3.55 到 3.65。
A rectangle 8 metres by 5 metres, each to the nearest metre.
一个 8 米 × 5 米的长方形，都精确到米。
The upper bounds are 8.5 and 5.5, so the largest area is 46.75.
上界是 8.5 和 5.5，所以最大面积是 46.75。
The lower bounds give 33.75.
下界给出 33.75。
But a quotient like length over width needs upper over lower: 8.5 over 4.5, about 1.9, because a smaller denominator pushes the answer up.
但长除以宽这样的商要用上界除以下界：8.5 ÷ 4.5，约 1.9，因为分母越小，结果越大。
Three of eight equal parts is three eighths.
八等份中的三份是 3/8。
As a decimal, 0.375 is 375 thousandths, which cancels back to three eighths; as a percentage, 37.5.
写成小数 0.375，即 375/1000，约分回到 3/8；写成百分数是 37.5%。
To compare three eighths with 0.4, write 0.375 and 0.400: three eighths is smaller.
比较 3/8 和 0.4，写成 0.375 和 0.400：3/8 较小。
Six hundredths, 0.06, is 3 fiftieths.
0.06 是百分之六，即 3/50。
Let x be 0.2727 recurring.
设 x = 0.2727…。
A hundred x has the same tail, so subtracting kills it: 99x is 27, and x is 3 elevenths.
100x 的尾巴相同，相减就消掉：99x = 27，x = 3/11。
For 0.1666 recurring the 1 does not repeat, so use two powers of ten: 100y minus 10y is 15, and y is one sixth.
对于 0.1666…，1 不循环，所以用两个 10 的幂：100y − 10y = 15，y = 1/6。
Start from 80.
从 80 开始。
A 15 percent increase multiplies by 1.15, giving 92.
增加 15% 乘以 1.15，得 92。
A 20 percent decrease multiplies by 0.8, giving 64.
减少 20% 乘以 0.8，得 64。
To undo a change, divide by the multiplier: 92 divided by 1.15 is the original 80.
要还原变化，就除以乘数：92 ÷ 1.15 回到原来的 80。
And a fraction is an operator too: three fifths of 40 is 24.
分数也是运算符：40 的 3/5 是 24。
How many five eighths fit in three quarters?
¾ 里有几个 5/8？
Write three quarters as six eighths.
把 ¾ 写成 6/8。
One group of five eighths fits, with one eighth left, which is one fifth of another group.
能放下一组 5/8，剩下 1/8，是另一组的五分之一。
So the answer is six fifths, the same as three quarters times eight fifths.
所以答案是 6/5，与 ¾ × 8/5 相同。
Never add denominators: a half plus a third is not two fifths.
分母不能相加：½ + ⅓ 不等于 2/5。
Share 84 yuan in the ratio 2 to 5.
把 84 元按 2:5 分。
Add the parts: seven equal parts, so each is 12.
先把份数相加：共七等份，每份 12。
The shares are 24 and 60.
两份分别是 24 和 60。
The first share is 2 sevenths of the whole, not 2 fifths; 2 fifths compares it with the other share.
第一份是总数的 2/7，不是 2/5；2/5 是它与另一份的比较。
Check: 24 plus 60 is 84.
检验：24 + 60 = 84。
A square tile is 0.4 metres wide.
一块正方形瓷砖宽 0.4 米。
Convert first: 0.4 metres is 40 centimetres, so the area is 40 times 40, 1600 square centimetres, the same as 0.16 times 10 000.
先换算：0.4 米是 40 厘米，所以面积是 40 × 40 = 1600 平方厘米，与 0.16 × 10 000 相同。
Volume cubes the factor: a litre is 1000 cubic centimetres.
体积要立方：1 升是 1000 立方厘米。
And 72 kilometres per hour is 72 000 metres over 3600 seconds: 20 metres per second.
72 千米/时是 72 000 米 ÷ 3600 秒：20 米/秒。
Both tiers use exact fractions, integer indices, standard form and simple error intervals.
两个层级都要用精确分数、整数指数、科学记数法和简单的误差区间。
Higher adds fractional indices and surds, recurring decimals as fractions, and bounds carried through a calculation.
高级卷另外有分数指数和根式、把循环小数化为分数，以及在计算中使用上下界。
Foundation answers keep exact fractions and multiples of pi instead.
基础卷则保留精确分数和 π 的倍数。
84 pencils and 126 pens go into identical packs, using everything.
84 支铅笔和 126 支钢笔装成相同的包，全部用完。
The number of packs must divide both, so the greatest is the HCF: 42.
包数必须同时整除两者，所以最多是最大公因数 42。
Each pack holds 2 pencils and 3 pens.
每包 2 支铅笔、3 支钢笔。
And a delivery charge of 18.60 shared by six buyers is exactly 3.10 each.
18.60 元运费由六人平分，每人正好 3.10 元。
A rectangle 8.0 by 3.0 centimetres, each to the nearest 0.1.
一个 8.0 × 3.0 厘米的长方形，都精确到 0.1。
Write both intervals.
写出两个区间。
The area runs from 23.4525, which is attained, up to 24.5525, which is not.
面积从 23.4525（可取到）到 24.5525（取不到）。
Length over width runs between 7.95 over 3.05 and 8.05 over 2.95, and neither end is reached, because each needs an excluded upper bound.
长除以宽在 7.95/3.05 和 8.05/2.95 之间，两端都取不到，因为每一端都需要一个被排除的上界。
Five terms to use in your reasons.
推理中要用的五个术语。
A prime factor: 60 is 2 squared times 3 times 5.
质因数：60 = 2² × 3 × 5。
A reciprocal: minus 4 and minus a quarter multiply to 1.
倒数：−4 和 −¼ 相乘得 1。
A surd: root 12 is 2 root 3.
根式：√12 = 2√3。
A recurring decimal: 0.27 recurring is 3 elevenths.
循环小数：0.27… = 3/11。
And significant figures start at the first nonzero digit: 0.004786 is 0.0048.
有效数字从第一个非零数字开始：0.004786 保留两位是 0.0048。
Check yourself.
自我检测。
The HCF of 72 and 90 is 18.
72 和 90 的最大公因数是 18。
Five sixths minus three quarters is one twelfth.
5/6 − 3/4 = 1/12。
27 to the two thirds is 9.
27^{2/3} = 9。
3.6 kilograms to 1 decimal place lies from 3.55 up to 3.65.
3.6 千克（一位小数）在 3.55 到 3.65 之间。
And if 20 percent off gives 64, the original was 64 divided by 0.8, which is 80.
打八折后是 64，原价是 64 ÷ 0.8 = 80。