Number
IGCSE Mathematics Topic 1 30:45 English narration · English + 中文 subtitles burned in
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Transcript
Two buses leave the same stop at nine o'clock.
两辆公交车在九点从同一个站出发。
One runs every twelve minutes, the other every eighteen.
一辆每十二分钟一班，另一辆每十八分钟一班。 它们下一次同时发车是在什么时候？
When do they next leave together? Not at thirty minutes. Not at two hundred and sixteen.
不是三十分钟后，也不是二百一十六分钟后， 而是三十六分钟后。
<slow>Thirty-six.</slow> And the fastest way to that answer is to break both numbers into primes — the one idea this whole topic is built on.
求出这个答案最快的方法，就是把两个数都分解成质因数—— 这正是整个专题赖以建立的核心观念。
This is Topic One: Number.
这是第一个专题：数。
It is the widest topic in the course, so this is a tour of the tools — primes, indices, standard form, percentages, ratio and accuracy.
它是整门课程中范围最广的专题， 所以这是一次工具的巡览——质数、指数、标准式、百分数、比例与精确度。
Parts marked Extended are only for the Extended papers.
标注 Extended 的部分只在扩展卷中考查。
Let's begin.
让我们开始吧。
First, the words.
先说术语。
Integers are the whole numbers, positive, negative and zero.
整数是所有的整数，包括正数、负数和零。
A prime has exactly two factors: one, and itself.
质数恰好有两个因数：一和它本身。
So two is prime — the only even one — and one is not prime, because it has only one factor.
所以二是质数——也是唯一的偶质数—— 而一不是质数，因为它只有一个因数。
Square numbers come from multiplying a number by itself; cube numbers from multiplying it three times.
平方数由一个数自乘得到， 立方数由自乘三次得到。
And a rational number can be written as a fraction of two integers. One that cannot — like pi, or the square root of two — is irrational.
有理数可以写成两个整数之比； 不能这样写的数——比如派，或者根号二——就是无理数。
The kinds of number fit inside one another like boxes.
各类数就像一个个盒子，层层套在一起。
The counting numbers — one, two, three and onwards — are the natural numbers.
用来数数的一、二、三……叫做自然数。
Add zero and the negatives, and you have the integers.
加上零和负数，就得到整数。
Allow any fraction of two integers, and you have the rational numbers.
再允许任意两个整数之比，就得到有理数。
Each set sits inside the next, so every natural number is also an integer, and every integer is also rational.
每一个集合都套在下一个里面：每个自然数也是整数，每个整数也是有理数。
Together with the irrational numbers they make up the real numbers, everything on the number line.
有理数和无理数合起来构成实数，也就是数轴上的所有数。
Irrational numbers are real too, but they sit outside the rationals, because no fraction can ever write them exactly.
无理数同样是实数，但它们待在有理数之外， 因为没有任何分数能把它们精确地写出来。
One more word while we are here: the reciprocal of a number is one divided by that number.
顺便再补一个词： 一个数的倒数就是一除以这个数。
The reciprocal of four is one quarter, and the reciprocal of two thirds is three halves.
四的倒数是四分之一， 三分之二的倒数是二分之三。
Two words the exam uses constantly, and students mix them up.
有两个词考试反复要用，而学生常常搞混。
A factor divides into a number exactly, leaving no remainder.
因数能整除一个数，没有余数。
The factors of eighteen are one, two, three, six, nine and eighteen.
十八的因数是一、二、三、六、九和十八。
They come in pairs that multiply back to eighteen: one times eighteen, two times nine, three times six.
它们成对出现，每一对乘回十八： 一乘十八、二乘九、三乘六。
Listing them in pairs is how you make sure none is missing.
按成对的方式列出来，才不会漏掉。
A multiple is that number times a whole number, so the multiples of six are six, twelve, eighteen, twenty-four and on forever.
倍数是这个数乘以一个整数，所以六的倍数是六、十二、十八、二十四，一直下去。
Factors are small and there are only a few of them; multiples are large and there is no end to them.
因数比原数小，个数有限；倍数比原数大，永远数不完。
Put two numbers together and you get two more words: a common factor divides into both of them, and a common multiple appears in both lists.
把两个数放在一起，又多出两个词：公因数能同时整除两个数， 公倍数同时出现在两串倍数里。
Now the payoff.
现在来看它的用处。
Break a number down with a factor tree until only primes are left.
用因数树把一个数分解，直到只剩下质数。
Twelve is two times two times three. Eighteen is two times three times three.
十二是二乘二乘三，十八是二乘三乘三。
Put them in a Venn diagram: shared primes in the middle, the rest outside.
把它们放进韦恩图： 共有的质因数放中间，其余放两边。
The highest common factor — HCF — is the product of the middle — two times three, which is six.
最大公因数是中间部分的乘积—— 二乘三，等于六。
The lowest common multiple — LCM — is the product of everything in the diagram — two times two times three times three, thirty-six.
最小公倍数是图中全部因数的乘积—— 二乘二乘三乘三，等于三十六。
So the buses meet again after thirty-six minutes.
所以两辆公交车在三十六分钟后再次同时发车。
Here is that method written out in full, on seventy-two.
现在把这个方法完整地写在七十二上。
Keep dividing by the smallest prime that fits.
不断除以能整除的最小质数。
Two fits, and seventy-two is two times thirty-six.
二可以，七十二是二乘三十六。
Thirty-six is two times eighteen.
三十六是二乘十八。
Then eighteen is two times nine.
十八又是二乘九。
Nine is not even, so move up to the next prime: nine is three times three.
九不是偶数，就换成下一个质数：九是三乘三。
Every branch has now ended on a prime, so stop.
现在每一条分支都停在质数上， 可以停下来了。
Count what you have — three twos and two threes — so seventy-two is two cubed times three squared. Always write the answer with powers like that.
数一数手里有什么——三个二和两个三—— 所以七十二等于二的三次方乘三的平方。
It is shorter, and the highest common factor questions that follow will want it in exactly that form.
答案一定要用幂写出来： 它更短，而且后面的最大公因数题正需要这种形式。
Now use prime factors to find a highest common factor and a lowest common multiple, on seventy-two and one hundred and twenty.
现在用质因数来求最大公因数和最小公倍数，题目是七十二和一百二十。
Write each one as a product of primes first. Seventy-two is two cubed times three squared, and one hundred and twenty is two cubed times three times five.
先把两个数都写成质因数的乘积：七十二是二的三次方乘三的平方， 一百二十是二的三次方乘三乘五。
For the highest common factor, take the lowest power of each prime that appears in both of them: two cubed times three, which is twenty-four.
求最大公因数时， 取每个共有质数的最低次幂：二的三次方乘三，等于二十四。
For the lowest common multiple, take the highest power of every prime that appears anywhere: two cubed times three squared times five, which is three hundred and sixty.
求最小公倍数时，取出现过的每个质数的最高次幂： 二的三次方乘三的平方乘五，等于三百六十。
Shared part only for the highest common factor; everything for the lowest common multiple.
最大公因数只取共有的部分，最小公倍数要全部都取。
Sets have their own small language, and Cambridge expects you to read it.
集合有自己的一套小语言，剑桥要求你能读懂它。
A set is a collection of things, written inside curly brackets.
集合就是一堆东西，写在花括号里。
Each thing in a set is called an element.
集合里的每一个东西叫做元素。
The rounded E symbol means "is an element of"; the same symbol with a line through it means "is not an element of"; and n of A means the number of elements in A.
圆圆的 E 形符号表示「属于」， 同一个符号加一道斜线表示「不属于」；括号里写 A、前面写 n， 表示集合 A 里元素的个数。
The universal set is everything being talked about in that question.
全集是这道题里所讨论的全部对象。
A with a dash after it is the complement of A, everything in the universal set that is not in A.
A 右上角加一撇是 A 的补集，也就是全集里不属于 A 的部分。
The empty set has no elements at all.
空集里一个元素都没有。
If every element of A is also in B, then A is a subset of B.
如果 A 的每个元素都在 B 里，那么 A 是 B 的子集。
Union means in A, or in B, or in both.
并集表示在 A 里、或在 B 里、或两个都在。
Intersection means in both at the same time.
交集表示同时在两个集合里。
Let's use that.
来用一下。
The universal set is the whole numbers from one to ten, A is the even numbers, and B is the multiples of three.
全集是一到十的整数，A 是偶数，B 是三的倍数。
First write the two lists. A is two, four, six, eight and ten. B is three, six and nine.
先写出两串数：A 是二、四、六、八、十；B 是三、六、九。
Six is in both, so six goes in the overlap, and the intersection of A and B is just six.
六同时在两串里，所以六放进重叠部分，A 和 B 的交集就只有六。
The union of A and B is everything inside either circle: two, three, four, six, eight, nine and ten.
A 和 B 的并集是两个圆里的全部：二、三、四、六、八、九、十。
Count them — seven elements — so n of A union B is seven.
数一数，一共七个元素，所以 A 并 B 的元素个数是七。
One, five and seven belong to neither list, so they sit inside the rectangle but outside both circles.
一、五、七都不属于这两串，所以它们在矩形里、两个圆外。
You may also meet a set written as a rule, such as all values of x from one to five.
你还可能遇到用条件写出的集合，比如「所有从一到五的 x」。
Powers and roots next.
接下来是幂和根。
A power tells you how many times to multiply a number by itself, and that small raised number has a name: the index.
幂表示把一个数自乘多少次，右上角那个小数字有个名字：指数。
Two to the fifth means five twos multiplied together, which is thirty-two.
二的五次方就是五个二相乘，等于三十二。
A square root undoes squaring, so the square root of one hundred and sixty-nine is thirteen, because thirteen squared is one hundred and sixty-nine.
平方根是平方的逆运算， 所以一百六十九的平方根是十三，因为十三的平方是一百六十九。
A cube root undoes cubing, so the cube root of eight is two.
立方根是立方的逆运算，所以八的立方根是二。
You are expected to know the squares from one to fifteen and their roots by heart, and the cubes of one, two, three, four, five and ten.
你要背下一到十五的平方数和它们的平方根， 以及一、二、三、四、五和十的立方。
That recall wins you time in the exam.
这些记忆在考场上能省下真金白银的时间。
So five squared times the cube root of eight is twenty-five times two, which is fifty.
所以五的平方乘八的立方根，就是二十五乘二，等于五十。
Powers have three rules, and you must know all three by heart.
幂有三条法则，你必须全部背熟。
Multiplying powers of the same base: add the indices.
同底数幂相乘：指数相加。
Dividing powers of the same base: subtract them.
同底数幂相除：指数相减。
A power raised to another power: multiply them.
幂的乘方：指数相乘。
Two special ones catch people out. Anything to the power zero is one. And a negative index means one over the positive version — it flips the number, it does not make it negative.
还有两个特殊情况常常让人出错：任何数的零次方都等于一； 负指数表示正指数的倒数——它把数取倒数，而不是把数变成负的。
Four quick examples on those laws.
用这几条法则做四道快题。
Two to the negative three, times two to the fourth: same base, so add the indices.
二的负三次方乘二的四次方：同底数，所以指数相加。
Negative three plus four is one, so the answer is just two.
负三加四等于一，所以答案就是二。
Next, two cubed all raised to the second power: multiply the indices, three times two is six, so it is two to the sixth, which is sixty-four.
再看二的三次方整体再平方：指数相乘， 三乘二等于六，所以是二的六次方，等于六十四。
Now two cubed divided by two to the fourth: subtract, three minus four is negative one, and a negative index means one over, so the answer is one half.
再看二的三次方除以二的四次方：指数相减，三减四等于负一， 而负指数表示取倒数，所以答案是二分之一。
Last one: seven to the negative two is one over seven squared, which is one over forty-nine.
最后一题：七的负二次方就是七的平方的倒数，等于四十九分之一。
Fractional indices are Extended only, and they are less frightening than they look.
分数指数只在扩展卷考，其实并不像看上去那么可怕。
A power of one over n is just the n-th root, so a power of a half is a square root and a power of a third is a cube root.
n 分之一次方就是 n 次根，所以二分之一次方是平方根，三分之一次方是立方根。
When the fraction has a top and a bottom, the bottom gives the root and the top number gives the power.
当分数有分子也有分母时，分母决定开几次根，分子决定乘几次方。
Take the root first and the power second, because that keeps the numbers small.
先开根、后乘方，这样数字才不会变大。
Eighty-one to the power a half is the square root of eighty-one, which is nine.
八十一的二分之一次方就是八十一的平方根，等于九。
And eight to the negative two thirds: the minus sign flips it, so it is one over eight to the two thirds; the cube root of eight is two, and two squared is four, so the answer is one quarter.
再看八的负三分之二次方：负号表示取倒数，所以是八的三分之二次方的倒数； 八的立方根是二，二的平方是四，所以答案是四分之一。
Very large and very small numbers get written in standard form.
很大和很小的数要写成标准式。
Standard form is one number between one and ten, times a power of ten.
标准式是一个介于一和十之间的数，乘以十的某次幂。
Watch the first part — it must be at least one, and less than ten.
注意前面那个数——它必须大于等于一并且小于十。
So four hundred and fifty thousand becomes four point five times ten to the fifth. And zero point zero zero zero six two becomes six point two times ten to the negative four.
所以四十五万写成四点五乘以十的五次方， 而零点零零零六二写成六点二乘以十的负四次方。
To multiply two numbers in standard form, multiply the front parts and add the powers.
两个标准式相乘时，把前面的数相乘，指数相加。
One worked example. Three times ten to the fifth, multiplied by two times ten to the negative two.
来看一道例题：三乘十的五次方，再乘二乘十的负二次方。
Split the job in two.
把工作分成两半。
Multiply the front numbers: three times two is six.
先乘前面的数：三乘二等于六。
Then add the powers: five plus negative two is three.
再把指数相加：五加负二等于三。
So the answer is six times ten to the third.
所以答案是六乘十的三次方。
The front number, six, is already between one and ten, so we are finished.
前面的数六已经在一到十之间，做完了。
One more thing to watch. If the front number came out as something like twelve, that is not standard form yet — you would rewrite twelve times ten to the third as one point two times ten to the fourth.
还有一点要注意：如果前面的数算出来是十二这样的数，那还不是标准式—— 要把十二乘十的三次方改写成一点二乘十的四次方。
Surds are Extended.
根式属于扩展内容。
A surd is a root that comes out irrational — the square root of five, for example.
根式就是开出来是无理数的根，比如五的平方根。
You leave it in that exact form instead of rounding, because rounding loses marks in an exact-answer question.
要保留这种精确形式，不要四舍五入，因为在要求精确答案的题里取近似会丢分。
Two rules do most of the work: root a times root b is root a b, and the root of a fraction is the root of the top over the root of the bottom.
两条法则包揽了大部分工作：根号 a 乘根号 b 等于根号 ab； 分数的根等于分子的根除以分母的根。
To simplify, take out the largest square factor.
化简时，把最大的平方因数提出来。
So the square root of twenty is the square root of four times five, which is root four times root five, which is two root five.
所以二十的平方根等于四乘五的平方根，也就是根号四乘根号五，等于二倍根号五。
Now the square root of two hundred minus the square root of thirty-two.
再看二百的平方根减三十二的平方根。
Two hundred is one hundred times two, giving ten root two; thirty-two is sixteen times two, giving four root two.
二百是一百乘二，得十倍根号二； 三十二是十六乘二，得四倍根号二。
Both are root twos now, so subtract them: six root two.
现在都是根号二了，相减得六倍根号二。
Rationalising the denominator means clearing the surd off the bottom of a fraction.
分母有理化，就是把根号从分数的分母里清掉。
The bottom of a fraction is called the denominator, and the exam wants no surd on the bottom of a final answer.
分数下面那部分叫分母， 而考试要求最终答案的分母里不能有根号。
The trick is to multiply the top and the bottom by something that clears it, which does not change the value, because you are really multiplying by one.
办法是把分子和分母同时乘上一个能清掉它的量； 这不会改变值，因为实际上是在乘一。
Ten over root five: multiply top and bottom by root five.
看十除以根号五：分子分母同乘根号五。
The bottom becomes five, the top becomes ten root five, and that cancels to two root five.
分母变成五，分子变成十倍根号五，约掉之后是二倍根号五。
Now a harder one: one over root three minus one.
再看难一点的：一除以根号三减一。
Multiply top and bottom by root three plus one.
分子分母同乘根号三加一。
The bottom becomes three minus one, which is two, so the answer is root three plus one over two.
分母变成三减一，也就是二，所以答案是根号三加一再除以二。
Now fractions, and the words come first.
现在讲分数，先讲名称。
The top of a fraction is the numerator and the bottom is the denominator.
分数上面那部分叫分子，下面那部分叫分母。
Three kinds come up in the exam.
考试里会出现三种分数。
A proper fraction has a numerator smaller than its denominator, like three quarters.
真分数的分子比分母小，比如四分之三。
An improper fraction has a numerator the same size or bigger, like seven over four.
假分数的分子等于或大于分母，比如四分之七。
And a mixed number is a whole number with a fraction beside it, like one and three quarters.
带分数是一个整数旁边带一个分数，比如一又四分之三。
Changing between the last two is just division: seven over four means seven divided by four, which is one remainder three, so it is one and three quarters.
后两者之间的转换其实就是除法：四分之七表示七除以四，商一余三， 所以等于一又四分之三。
Last, simplest form. Divide the top and the bottom by their highest common factor: eighteen over twenty-four, both divided by six, is three quarters.
最后是最简形式：把分子和分母同时除以它们的最大公因数， 二十四分之十八上下同除以六，等于四分之三。
The same amount can be written three ways, and the exam moves between them constantly.
同一个量可以用三种方式表示，而考试会不断在它们之间转换。
To convert, go through the decimal.
转换时以小数为桥梁。
A fraction becomes a decimal by dividing the top by the bottom.
分数化小数：用分子除以分母。
A decimal becomes a percentage by multiplying by one hundred.
小数化百分数：乘以一百。
And to put a mixed list in order, turn every one of them into a decimal first.
要把混在一起的一列数排序， 先把每一个都化成小数。
Comparing a fraction with a percentage directly is where mistakes happen.
直接拿分数和百分数比较，正是出错的地方。
Here is the conversion table from the handout, and it is worth learning as four one-line moves.
这是讲义里的换算表，值得当成四步一行的口诀背下来。
A fraction becomes a decimal when you divide the top by the bottom: three eighths is three divided by eight, which is nought point three seven five.
分数化小数：用分子除以分母，八分之三就是三除以八，等于零点三七五。
A decimal becomes a percentage when you multiply by one hundred, so nought point nought seven is seven per cent.
小数化百分数：乘以一百，所以零点零七是百分之七。
A percentage becomes a fraction when you put it over one hundred and simplify, so seven per cent is seven hundredths.
百分数化分数：写成一百分之几再约简，所以百分之七是一百分之七。
And a percentage goes back to a decimal when you divide by one hundred, so thirty-four per cent is nought point three four.
百分数化回小数：除以一百，所以百分之三十四是零点三四。
Recurring decimals are Extended.
循环小数属于扩展内容。
A dot over a digit means that digit repeats for ever, so nought point one with a dot on the seven is nought point one seven seven seven, on and on.
数字上面的点表示这个数字无限重复， 所以零点一、七上面加点，就是零点一七七七，一直循环下去。
Dots on the first and last digit of a group mean the whole group repeats.
一组数字首尾各加一点，表示整组重复。
To turn one into a fraction, line the repeating tails up and subtract.
要把它化成分数， 就让循环的尾巴对齐，然后相减。
Call the number x.
把这个数记作 x。
Only the seven repeats, so use ten x and one hundred x: one hundred x is seventeen point seven seven seven, and ten x is one point seven seven seven.
只有七在循环， 所以用十倍 x 和一百倍 x：一百倍 x 是十七点七七七，十倍 x 是一点七七七。
Subtract the two lines and the endless tails cancel exactly, leaving ninety x equals sixteen.
两行相减，无限长的尾巴刚好抵消，剩下九十倍 x 等于十六。
So x is sixteen over ninety, which cancels to eight over forty-five.
所以 x 等于九十分之十六，约简后是四十五分之八。
Order of operations now.
现在讲运算顺序。
Brackets first, then indices — that means powers and roots — then multiply and divide working left to right, then add and subtract, also left to right.
先括号，再指数——也就是乘方和开方—— 然后从左到右做乘除，最后从左到右做加减。
Most students learn it as BIDMAS.
多数学生把它记成 BIDMAS。
Try minus six times minus three, plus seven times two.
试一试：负六乘负三，加七乘二。
The multiplications go first: minus six times minus three is eighteen, and seven times two is fourteen.
先做乘法：负六乘负三等于十八，七乘二等于十四。
Only then add, giving thirty-two.
然后才相加，得三十二。
Negative numbers have three habits of their own.
负数另有三个习惯要记牢。
Adding a negative is the same as subtracting, so five add negative three is two.
加一个负数等于减， 所以五加负三等于二。
Subtracting a negative is the same as adding, so five take away negative three is eight.
减一个负数等于加，所以五减负三等于八。
And when multiplying or dividing, same signs give a positive, different signs give a negative — which is why a temperature change from minus five degrees to three degrees is a rise of eight degrees.
做乘除时，同号得正，异号得负—— 所以温度从负五度升到三度，是上升了八度。
Calculating with fractions.
分数运算。
To multiply fractions, multiply the tops and multiply the bottoms — no common denominator needed.
分数相乘：分子乘分子，分母乘分母，不需要通分。
Two thirds times four fifths is eight fifteenths.
三分之二乘五分之四等于十五分之八。
To divide, turn the second fraction upside down and multiply instead: two thirds divided by four fifths becomes two thirds times five quarters, which is ten twelfths, and that cancels to five sixths.
分数相除：把第二个分数上下翻过来再相乘， 三分之二除以五分之四就变成三分之二乘四分之五，等于十二分之十，约简后是六分之五。
To add or subtract you do need a common denominator, and the lowest common multiple of the bottoms is the one to use.
加减法就必须通分了，取分母的最小公倍数作公分母。
Try one and seven fifteenths, take away four fifths.
试试一又十五分之七减五分之四。
Change the mixed number to twenty-two fifteenths, change four fifths to twelve fifteenths, subtract to get ten fifteenths, and cancel to two thirds.
先把带分数化成十五分之二十二， 再把五分之四化成十五分之十二，相减得十五分之十，约简后是三分之二。
Ordering.
排序。
Six symbols compare two numbers by magnitude — by size: equal to, not equal to, greater than, less than, greater than or equal to, and less than or equal to.
有六个符号用来比较两个数的大小：等于、不等于、大于、小于、 大于或等于、小于或等于。
The two with a line under them include the value itself, and that will matter in the bounds questions later.
带横线的两个把这个值本身也包括进去， 这在后面的界限问题里很重要。
To put a mixed list in order, change every value to a decimal first — that is the whole method.
要把混在一起的一列数排序， 先把每一个都化成小数——整套方法就是这一句。
Put thirty-four per cent, one third and three tenths in order. As decimals, thirty-four per cent is nought point three four, one third is nought point three three three, and three tenths is nought point three.
把百分之三十四、三分之一和十分之三排序：化成小数后， 百分之三十四是零点三四，三分之一是零点三三三，十分之三是零点三。
So, smallest first, the order is three tenths, then one third, then thirty-four per cent.
所以从小到大是：十分之三、三分之一、百分之三十四。
Percentages carry a lot of marks, so we build them up carefully.
百分数在考试里占分很多，所以我们一步一步搭起来。
To find a percentage of an amount, change the percentage into a decimal and multiply.
求一个量的百分之几：把百分数化成小数再相乘。
Fifteen per cent of eighty dollars is nought point one five times eighty, which is twelve dollars.
八十美元的百分之十五就是零点一五乘八十，等于十二美元。
To write one amount as a percentage of another, divide, then multiply by one hundred.
把一个量写成另一个量的百分之几：先相除，再乘一百。
A score of eighteen out of twenty-five is eighteen over twenty-five times one hundred, which is seventy-two per cent.
二十五分里得十八分，就是二十五分之十八乘一百，等于百分之七十二。
Now a change.
再看变化。
A price rises from forty dollars to fifty dollars.
价格从四十美元涨到五十美元。
The change is ten dollars, and it is measured against the original forty, so ten over forty times one hundred is a twenty-five per cent increase.
变化量是十美元， 而它要拿原来的四十去比，所以四十分之十乘一百，是上涨百分之二十五。
Against the fifty you would get twenty per cent, and that is the wrong answer.
拿五十去比会得到百分之二十，那是错的。
Percentage change is the change divided by the original amount, times one hundred.
百分数变化等于变化量除以原来的量，再乘以一百。
Note that word — original. Dividing by the new amount is the classic error.
注意「原来的」这个词—— 用变化后的量去除，是最典型的错误。
But the fast way to apply a change is a multiplier.
而应用变化最快的方法是使用乘数。
An increase of fifteen percent means multiplying by one point one five.
增加百分之十五，就是乘以一点一五。
A decrease works the same way: down twenty percent means multiplying by zero point eight.
减少也一样： 减少百分之二十，就是乘以零点八。
One multiplication, no adding or subtracting.
一次乘法，不用加也不用减。
Two Extended tools now.
现在是两个扩展内容的工具。
A reverse percentage gives you the amount after the change and asks for the amount before it.
逆百分数问题给你变化之后的量，要求变化之前的量。
You divide by the multiplier — you do not subtract the percentage back off.
做法是除以乘数——而不是把那个百分数再减回去。
A price of sixty pounds after a twenty percent discount was sixty divided by zero point eight, which is seventy-five.
打八折之后是六十英镑，原价就是六十除以零点八，等于七十五。
Compound interest is the same multiplier, applied again and again: the starting amount times the multiplier to the power of the number of years.
复利用的是同一个乘数，反复相乘：本金乘以乘数的年数次方。
Interest is money paid for borrowing, or earned for saving, and the amount you start with is called the principal.
利息是借钱要付、存钱能得的钱，而一开始的那笔钱叫本金。
There are two kinds and the exam expects you to tell them apart.
利息有两种，考试要求你能分清。
Simple interest pays the same amount every year, worked out on the principal only: the principal times the rate times the number of years, all over one hundred.
单利每年付同样多，只按本金计算： 本金乘年利率乘年数，再除以一百。
Compound interest adds each year's interest onto the total, so the next year earns interest on a bigger amount.
复利把每年的利息加回本金， 所以下一年是在更大的总额上生息。
Its formula is the principal, times one plus the rate over a hundred, all raised to the number of years.
它的公式是： 本金乘以「一加百分率」，整体再取年数次方。
Try five hundred dollars at four per cent compound interest for three years.
试试五百美元、年利率百分之四的复利，存三年。
The multiplier is one point nought four, so five hundred times one point nought four cubed is five hundred and sixty-two dollars and forty-three cents.
乘数是一点零四， 所以五百乘一点零四的三次方，等于五百六十二美元四十三美分。
Two Extended finishes on percentages.
百分数还有两个扩展内容作结尾。
First, the handout's reverse question: a coat costs sixty dollars after a twenty per cent increase, and we want the original price.
第一，讲义里的逆百分数题： 一件外套在涨价百分之二十之后卖六十美元，要求原价。
Sixty dollars is not one hundred per cent of anything; it is one hundred and twenty per cent of the original.
六十美元不是任何东西的百分之一百，它是原价的百分之一百二十。
So divide by the multiplier one point two: sixty divided by one point two is fifty dollars.
所以要除以乘数一点二：六十除以一点二等于五十美元。
Taking twenty per cent off sixty would have given forty-eight, and that is simply wrong.
从六十里减掉百分之二十会得到四十八，那是错的。
Second, exponential growth and decay.
第二，指数增长与衰减。
Exponential decay is the same compound formula with a multiplier below one, and when something like a car loses value each year we call it depreciation.
指数衰减用的是同一个复利公式，只是乘数小于一； 像汽车这样每年贬值，我们叫折旧。
A car worth twenty thousand dollars loses fifteen per cent a year, so the multiplier is nought point eight five, and after four years it is worth about ten thousand four hundred and forty dollars.
一辆价值两万美元的车每年掉价百分之十五，乘数是零点八五， 四年后大约值一万零四百四十美元。
Ratio.
比。
A ratio compares quantities and is written with a colon between them.
比用来比较几个量，中间用冒号隔开。
Simplify a ratio exactly like a fraction, by dividing every part by their highest common factor: twenty to thirty to forty divides by ten, giving two to three to four.
化简比就跟化简分数一样， 把每一份都除以它们的最大公因数：二十比三十比四十，同除以十，得二比三比四。
Now share forty-eight dollars in the ratio three to five.
现在把四十八美元按三比五分配。
Add the parts first: three plus five makes eight parts.
先把份数相加：三加五，一共八份。
Then find one part: forty-eight divided by eight is six dollars.
再求一份是多少：四十八除以八，等于六美元。
Then multiply out — three parts and five parts — giving eighteen dollars and thirty dollars.
然后分别乘出来——三份和五份——得到十八美元和三十美元。
Check by adding: eighteen and thirty do make forty-eight.
加起来验算：十八加三十，正好是四十八。
Proportion is the same idea with one step — use it for recipes, map scales, and best value. If three pens cost one dollar eighty, then one pen costs sixty cents, so seven pens cost four dollars twenty.
比例是同样的想法，只多一步：如果三支笔卖一点八美元， 那一支笔六十美分，所以七支笔卖四点二美元。
To share an amount in a given ratio, add the parts to get the total number of shares, divide the amount by that total to find one share, then multiply out.
按给定的比分配一个量：先把各份相加得到总份数，用总量除以总份数求出一份， 再分别乘出来。
Share ninety in the ratio two to three to four: nine shares, so one share is ten, giving twenty, thirty and forty.
把九十按二比三比四分配：共九份，所以一份是十， 分别得到二十、三十和四十。
In direct proportion, doubling one doubles the other, and their quotient stays fixed.
成正比时，一个量加倍，另一个也加倍，两者的商不变。
In inverse proportion, doubling one halves the other, and their product stays fixed.
成反比时，一个量加倍，另一个减半，两者的积不变。
Rates.
比率。
A rate compares two quantities measured in different units — a price per kilogram, or a distance per hour.
比率比较的是两个单位不同的量——比如每公斤多少钱，或者每小时多少路程。
The one you must know is average speed.
你必须掌握的是平均速度。
Average speed is total distance divided by total time, and the triangle helps: cover the letter you want, and the other two tell you what to do.
平均速度等于总路程除以总时间， 三角图很好用：把要求的那个字母盖住，剩下两个就告诉你该怎么算。
A cyclist rides forty-five kilometres in three hours forty-five minutes.
一位骑车的人在三小时四十五分钟里骑了四十五公里。
Change the time into hours first, and this is exactly where marks get lost: three hours forty-five minutes is three point seven five hours, not three point four five.
先把时间化成小时，而这里正是最容易丢分的地方： 三小时四十五分钟是三点七五小时，不是三点四五。
Then forty-five divided by three point seven five is twelve kilometres per hour.
然后四十五除以三点七五，等于每小时十二公里。
Other rates work the same way. Density is mass divided by volume.
其他比率也一样： 密度等于质量除以体积。
Flow rate, fuel consumption and population density all follow the pattern, and if a rate needs a special formula, like pressure, the question will give it to you.
流量、油耗和人口密度都遵循同一个模式； 如果某个比率需要特殊公式，比如压强，题目会直接给你。
Two ways to round, and they are not the same.
有两种取近似的方式，而它们并不相同。
Decimal places count digits after the point.
小数位数数的是小数点后面的数字。
Significant figures start counting from the first non-zero digit — so in zero point zero zero four seven three, the first significant figure is the four.
有效数字从第一个非零数字开始数——所以在零点零零四七三中， 第一个有效数字是四。
To estimate an answer, round every number to one significant figure and work it out in your head.
要估算结果，就把每个数都取到一位有效数字，然后心算。
The exam awards marks for a sensible estimate, so never leave that blank.
考试对合理的估算是给分的，所以这一空绝不要留白。
Some practice with the handout's own numbers.
用讲义里的数字练一练。
The rule is always the same: look at the next digit — if it is five or more, round up; if it is less, round down.
规则永远一样：看下一位数字—— 五或更大就进位，小于五就舍去。
Three point one four one five nine to two decimal places: the next digit is one, so round down to three point one four.
三点一四一五九保留两位小数： 下一位是一，所以舍去，得三点一四。
Now significant figures.
再看有效数字。
Five thousand seven hundred and sixty-four to one significant figure: the first significant figure is the five, the next digit is seven, so round up to six thousand, and keep the place value with zeros.
五千七百六十四保留一位有效数字：第一位有效数字是五，下一位是七， 所以进位成六千，并用零把位值补齐。
Then nought point nought nought four nought six seven to two significant figures: leading zeros never count, so the first two significant figures are the four and the nought, the next digit is six, and it rounds to nought point nought nought four one.
再看零点零零四零六七保留两位有效数字：前面的零一律不算， 所以头两位有效数字是四和零，下一位是六，所以得零点零零四一。
Estimation is a marked skill of its own, and it is quick.
估算本身就是一项给分的能力，而且做起来很快。
To estimate an answer, round every number to one significant figure, then work it out in your head.
要估算结果， 先把每个数都取到一位有效数字，然后心算。
Estimate forty-one point three, over nine point seven nine times nought point seven six five.
估算四十一点三除以「九点七九乘零点七六五」。
Forty-one point three becomes forty, nine point seven nine becomes ten, and nought point seven six five becomes nought point eight.
四十一点三变成四十，九点七九变成十，零点七六五变成零点八。
Ten times nought point eight is eight, so the estimate is forty over eight, which is five.
十乘零点八等于八，所以估算结果是四十除以八，等于五。
Never leave an estimation part blank: the marks are for rounding sensibly, not for a perfect answer.
估算这一空绝不要留白：分数是给「合理取近似」的，不是给完美答案的。
Last tool, and it is Extended — limits of accuracy.
最后一个工具，属于扩展内容。
Any rounded measurement is only accurate to half a unit either way.
任何取过近似的测量值，其精确度都只到半个单位。
A length given as twelve centimetres, to the nearest centimetre, is really anywhere from eleven point five to twelve point five.
精确到厘米的十二厘米，实际上在十一点五到十二点五之间。
And when you combine bounds, think about what makes the answer biggest.
在把上下界组合起来时，要想清楚什么会让结果最大。
For the largest area, use both upper bounds. But for the largest result of a division, use the upper bound on top and the lower bound underneath.
求最大面积时，两个都取上界； 但求除法的最大结果时，分子取上界，分母取下界。
Bounds, with the handout's numbers.
用讲义里的数字来看界限。
A height is measured as six hundred and thirty-five metres, correct to the nearest metre.
一个高度测得是六百三十五米，精确到米。
Half a metre either side gives the band: six hundred and thirty-four point five is less than or equal to h, which is less than six hundred and thirty-five point five.
两边各留半米，就得到这个范围：六百三十四点五小于或等于 h， h 小于六百三十五点五。
Look closely at those two symbols. The lower bound is included, because it does round up to six hundred and thirty-five. The upper bound is not included, because it would round up to six hundred and thirty-six instead.
仔细看这两个符号：下界是包含在内的， 因为它确实会四舍五入成六百三十五；上界不包含在内， 因为它会进位成六百三十六。
So the lower bound is six hundred and thirty-four point five metres, and the upper bound is six hundred and thirty-five point five metres.
所以下界是六百三十四点五米， 上界是六百三十五点五米。
Combining bounds is Extended, and the whole trick is to ask what makes the answer biggest.
界限的组合运算属于扩展内容，全部诀窍就是想清楚：什么会让结果最大。
A rectangle measures eight centimetres by five, each to the nearest centimetre.
一个长方形是八厘米乘五厘米，两边都精确到厘米。
The largest possible area uses both upper bounds: eight point five times five point five is forty-six point seven five square centimetres.
最大可能面积要两边都取上界：八点五乘五点五，等于四十六点七五平方厘米。
The smallest area uses both lower bounds: seven point five times four point five is thirty-three point seven five.
最小面积则两边都取下界：七点五乘四点五，等于三十三点七五。
And for a divided quantity like speed, the largest possible speed comes from the largest distance divided by the smallest time — opposite bounds, not both upper.
而像速度这样的除法，最大速度来自最大的路程除以最小的时间—— 要取相反的界，而不是两边都取上界。
Time first — easy marks that get thrown away.
先讲时间——这里的分最容易白丢。
Sixty seconds make a minute, sixty minutes an hour, twenty-four hours a day, and take a year as three hundred and sixty-five days.
六十秒是一分钟，六十分钟是一小时， 二十四小时是一天，一年按三百六十五天算。
The twenty-four hour clock writes a time as four digits with no a.m. or p.m., so a quarter past three in the afternoon is fifteen fifteen.
二十四小时制用四位数字表示时刻，不写上午下午， 所以下午三点十五分写成十五点十五。
A film starts at nineteen thirty-five and lasts seventy minutes.
一部电影十九点三十五开始，放七十分钟。
Seventy minutes is one hour and ten minutes, so add the hour to get twenty thirty-five, then add the ten minutes to get twenty forty-five.
七十分钟是一小时十分钟， 先加一小时得二十点三十五，再加十分钟得二十点四十五。
Always add the hours first, then the minutes.
永远先加小时，再加分钟。
For a time zone question, add or subtract the difference between the two places.
遇到时区问题，就把两地的时差加上或减去。
Now money.
再讲货币。
Money answers are decimals given to two decimal places, so four point eight dollars is written four dollars eighty — that last zero is not optional.
货币答案是保留两位小数的小数， 所以四点八美元要写成四美元八十美分——最后那个零不能省。
Currency conversion: an exchange rate is just a multiplier.
而汇率就是一个乘数。
At one dollar to nought point nine two euros, a hundred and fifty dollars is a hundred and fifty times nought point nine two, which is a hundred and thirty-eight euros.
一美元兑零点九二欧元时， 一百五十美元就是一百五十乘零点九二，等于一百三十八欧元。
To come back the other way you divide by the same rate: a hundred and thirty-eight divided by nought point nine two is a hundred and fifty dollars again.
要换回去，就用同一个汇率去除：一百三十八除以零点九二，又回到一百五十美元。
Two last things: the calculator, and the tips that save the most marks.
最后两件事：计算器，以及最能保住分数的几条提醒。
Do not round part-way through a calculation — keep the full value on the display and round only the final answer, to whatever accuracy the question asks for.
不要在中途取近似——把完整的数留在屏幕上，只对最后的答案取近似， 并按题目要求的精确度来取。
Enter time as a decimal of an hour: two hours thirty minutes is two point five hours, never two point three zero.
时间要输入成小时的小数形式： 两小时三十分钟是二点五小时，绝不是二点三零。
Read the display in context, too: in money, four point eight means four dollars eighty; in time, three point two five hours means three hours fifteen minutes.
也要结合情境读屏幕上的数：在货币里，四点八表示四美元八十美分； 在时间里，三点二五小时表示三小时十五分钟。
Then the tips. Always follow BIDMAS, and remember that a negative times a negative is positive.
再说几条提醒： 永远按 BIDMAS 的顺序算，并记住负数乘负数得正数。
And in standard form the number in front must be between one and ten, with a negative power of ten for a small number like nought point nought nought nought four.
标准式里前面那个数必须在一到十之间； 像零点零零零四这样的小数，十的指数是负的。
One question to finish.
最后来看一道题。
After a discount of thirty percent, a jacket costs forty-nine pounds.
打七折之后，一件夹克卖四十九英镑。
What was the price before the discount?
打折前的价格是多少？
Pause here and try it.
先暂停，自己试一试。
Spot the type first — you are given the amount after the change, so this is a reverse percentage.
先判断题型——题目给的是变化之后的量，所以这是逆百分数问题。
The multiplier for down thirty percent is zero point seven.
减少百分之三十对应的乘数是零点七。
Now divide, do not subtract.
现在要除，而不是减。
So the price before the discount was forty-nine divided by zero point seven, which is seventy pounds.
所以打折前的价格是四十九除以零点七，等于七十英镑。
Check it: seventy times zero point seven really is forty-nine.
验算一下：七十乘以零点七，确实是四十九。
Three habits that save marks all year.
三个全年都能保住分数的习惯。
First, for percentage change, divide by the original amount, never the new one.
第一，求百分数变化时，要除以原来的量， 绝不是变化后的量。
Second, a reverse percentage divides by the multiplier — subtracting the percentage back gives the wrong answer every time.
第二，逆百分数问题要除以乘数—— 把百分数减回去每一次都会算错。
Third, count significant figures from the first non-zero digit, not from the decimal point.
第三，有效数字从第一个非零数字开始数， 而不是从小数点开始数。
Get those three, and Topic One is yours.
做到这三点，第一个专题就是你的了。