BODMAS and Simplification

The Order of Operations: VBODMAS

A simplification question gives you a long line of numbers and signs and four options that sit close together. The arithmetic itself is never hard. What decides the answer is the order in which you do the steps, and that order has a name: VBODMAS.

Powers and roots ($5^2$, $\sqrt{49}$) are worked out right after the brackets, before "of" and the four basic operations.

The rule people forget. D comes before M in the word, but division does not beat multiplication. They have equal rank, so you take whichever comes first as you read from left to right. Addition and subtraction work the same way.

$$60 \div 5 \times 3 = 12 \times 3 = 36$$

Doing the multiplication first gives 60 ÷ 15 = 4, which is wrong. In the same way, 20 − 8 + 3 = 12 + 3 = 15, not 20 − 11 = 9.

"of" is not the same as ×. Both mean multiply, but "of" is done earlier.

$$36 \div 4 \text{ of } 3 = 36 \div 12 = 3 \qquad 36 \div 4 \times 3 = 9 \times 3 = 27$$

The same three numbers give 3 with "of" and 27 with ×, and SSC puts both values in the options.

A full run. Simplify 18 + 24 ÷ 3 of 2 − 5 × 2.

• "of" first: 3 of 2 = 6
• Division and multiplication, left to right: 24 ÷ 6 = 4 and 5 × 2 = 10
• Addition and subtraction, left to right: 18 + 4 − 10 = 12

Key Point: Bar, brackets, powers, "of", then ÷ and × from left to right, then + and − from left to right. Most wrong options are made by breaking the left-to-right rule.

Brackets, the Vinculum and Signs

Which bracket first. Work from the inside out: the vinculum (bar) first, then the round brackets ( ), then the curly brackets { }, and the square brackets [ ] last. A vinculum is just a bracket drawn as a line, so $\overline{15 - 6}$ means (15 − 6).

$$40 - \overline{15 - 6} = 40 - 9 = 31$$

Ignore the bar and you get 40 − 15 − 6 = 19, which is a ready-made wrong option.

A nested example. Simplify $50 - [24 - \{18 - (11 - \overline{6 - 2})\}]$.

• Bar: 6 − 2 = 4
• Round bracket: 11 − 4 = 7
• Curly bracket: 18 − 7 = 11
• Square bracket: 24 − 11 = 13
• Last step: 50 − 13 = 37

Write each layer's value on rough paper as you go. Trying to open all the brackets at once is where the sign mistakes come from.

A minus sign in front of a bracket. When you remove the bracket, every sign inside flips: 30 − (12 − 5) = 30 − 12 + 5 = 23. If you work out the bracket first (30 − 7 = 23) you never have to think about flipping, which is the safer habit.

Signs in × and ÷.

Division follows the same table: (−18) ÷ (−3) = 6 and 18 ÷ (−3) = −6.

Powers of negative numbers. An even power of a negative number is positive and an odd power is negative: $(-2)^4 = 16$ and $(-2)^3 = -8$. Watch the brackets. $-3^2$ means $-(3^2) = -9$, because the power belongs to the 3 alone, while $(-3)^2 = 9$.

Putting it together. For (−12) ÷ (−3) × (−2) + 5, go left to right: (−12) ÷ (−3) = 4, then 4 × (−2) = −8, then −8 + 5 = −3.

A fraction bar is a bracket too. In $\frac{12 + 8}{4 \times 5}$ the top and the bottom are each worked out on their own first, giving $\frac{20}{20} = 1$.

The Traps in Long Expressions

SSC builds its wrong options out of the same few slips. Once you can spot them, most BODMAS questions take under a minute.

Two facts sit behind the first two rows. A chain of divisions is the same as dividing by the product, so a ÷ b ÷ c = a ÷ (b × c). A chain of subtractions is the same as subtracting the sum, so a − b − c = a − (b + c).

Fractions inside BODMAS. A fraction such as $\frac{3}{4}$ is one number, so the order rules never split it up. Simplify

$$\frac{5}{6} \text{ of } \frac{3}{10} \div \frac{1}{4} + 2 \times \frac{3}{8} - \frac{1}{2}$$

• "of" first: $\frac{5}{6} \times \frac{3}{10} = \frac{1}{4}$
• Division: $\frac{1}{4} \div \frac{1}{4} = 1$
• Multiplication: $2 \times \frac{3}{8} = \frac{3}{4}$
• Last: $1 + \frac{3}{4} - \frac{1}{2} = \frac{5}{4} = 1\frac{1}{4}$

Mixed numbers and the rest of the fraction work are in the fractions section of this chapter.

A fraction made of two expressions. Simplify

$$\frac{7 \times 6 - 4 \times 3 \div 2}{48 \div 4 \text{ of } 2 + 3}$$

On top, 7 × 6 = 42 and 4 × 3 ÷ 2 = 12 ÷ 2 = 6, so the top is 36. On the bottom, 4 of 2 = 8, 48 ÷ 8 = 6, and 6 + 3 = 9. The value is 36 ÷ 9 = 4. If "of" is read as an ordinary ×, the bottom turns into 48 ÷ 4 × 2 + 3 = 27, and the wrong value $\frac{36}{27} = \frac{4}{3}$ is usually one of the choices.

[SSC Tip] Glance at the options before you start. If they are far apart, a rough estimate is enough to pick one. If two are close, one of them is almost always built from a slip in the table above, so do the risky step slowly.

Decimals and Fractions: Conversions to Know by Heart

Decimal to fraction. Write the digits after the point over 10, 100, 1000 and so on (one zero for each decimal place), then reduce.

$$0.45 = \frac{45}{100} = \frac{9}{20} \qquad 2.375 = \frac{2375}{1000} = \frac{19}{8}$$

Fraction to decimal. Either divide, or make the bottom a power of 10: $\frac{7}{25} = \frac{28}{100} = 0.28$ and $\frac{3}{8} = \frac{375}{1000} = 0.375$.

Most simplification sums get much shorter once you swap a decimal for its fraction. These are worth learning by heart:

Multiples follow straight away: $\frac{3}{8}$ = 0.375, $\frac{5}{8}$ = 0.625, $\frac{7}{8}$ = 0.875 and $\frac{3}{16}$ = 0.1875. So 0.875 × 64 is just $\frac{7}{8}$ × 64 = 56, and 0.125 × 0.64 is 0.64 ÷ 8 = 0.08.

Comparing decimals. Give them all the same number of decimal places by adding zeros, then compare them like whole numbers. To order 0.07, 0.7, 0.077 and 0.707, write 0.070, 0.700, 0.077 and 0.707. So 0.07 < 0.077 < 0.7 < 0.707.

Decimal place value. In 63.2048 the digits stand for 6 tens, 3 ones, 2 tenths, 0 hundredths, 4 thousandths and 8 ten-thousandths:

$$63.2048 = 6 \times 10 + 3 \times 1 + \frac{2}{10} + \frac{4}{1000} + \frac{8}{10000}$$

SSC turns this round. It writes something like 63.2048 = 6 × A + 3 × B + 2 ÷ C + 4 ÷ D + 8 ÷ E and asks for some combination of A to E. Match each digit to its place and the letters fall out: A = 10, B = 1, C = 10, D = 1000 and E = 10000. A zero digit (here the hundredths) simply has no term.

Decimal Arithmetic Without Slips

Adding and subtracting. Line up the decimal points and fill the gaps with zeros. 12.4 + 3.086 + 0.97 becomes 12.400 + 3.086 + 0.970 = 16.456. For 7 − 2.358, write 7.000 − 2.358 = 4.642.

Multiplying. Ignore the points, multiply the whole numbers, then count the total number of decimal places in the question and put that many in the answer.

• 0.6 × 0.05: 6 × 5 = 30, and there are 1 + 2 = 3 places, so 0.030 = 0.03.
• 0.12 × 0.3 × 0.02: 12 × 3 × 2 = 72, and there are 2 + 1 + 2 = 5 places, so 0.00072.
• $(0.03)^2$: 3 × 3 = 9 with 4 places, so 0.0009. Squaring a decimal doubles its places.

Dividing. Move the decimal point in both numbers by the same number of places until the divisor is a whole number. Moving both by the same amount does not change the answer.

• 0.0084 ÷ 0.12 = 0.84 ÷ 12 = 0.07
• 4.5 ÷ 0.009 = 4500 ÷ 9 = 500

Powers of 10. Multiplying by $10^k$ moves the point $k$ places to the right, and dividing moves it $k$ places to the left: 3.47 × 1000 = 3470 and 52 ÷ $10^4$ = 0.0052.

Clearing decimals from a fraction. When decimals sit on the top and bottom of a fraction, work out each part, then multiply both by the same power of 10.

$$\frac{0.2 \times 0.2 + 0.01}{0.1 \times 0.1 + 0.02} = \frac{0.04 + 0.01}{0.01 + 0.02} = \frac{0.05}{0.03} = \frac{5}{3}$$

Two decimals in a ratio. If $0.5a = 0.02b$, then $\frac{a}{b} = \frac{0.02}{0.5} = \frac{2}{50} = \frac{1}{25}$. So $b = 25a$, and a question like "find $\frac{b - a}{b + a}$" becomes $\frac{24a}{26a} = \frac{12}{13}$. Always work out which letter is bigger before you substitute. Flipping the ratio is the trap here.

[SSC Tip] After a decimal multiplication or division, do a size check. 0.0084 ÷ 0.12 is roughly 0.01 ÷ 0.1 = 0.1, so 0.07 fits, while 0.7 and 0.007 do not. Options in decimal questions often differ only in where the point sits, so this check alone picks the answer.

Percentages Inside Simplification

In simplification, a percentage is just a fraction with 100 underneath: x% of y = $\frac{x}{100} \times y$. Treat "x% of y" as a single value and work it out early, the same way as "of".

Use the fraction, not the percent. With the fraction table above, many percentages become one division:

• 12.5% of 640 = $\frac{1}{8}$ × 640 = 80
• 37.5% of 96 = $\frac{3}{8}$ × 96 = 36
• $16\frac{2}{3}$% of 162 = $\frac{1}{6}$ × 162 = 27
• 62.5% of 56 = $\frac{5}{8}$ × 56 = 35

Swap the numbers. x% of y = y% of x, because both equal $\frac{xy}{100}$. So 4% of 75 = 75% of 4 = 3, and 28% of 25 = 25% of 28 = 7. When one way round is awkward, swap.

Break it up. 35% of 460 is three lots of 10% plus 5%: 46 × 3 + 23 = 161. And 18% of 350 = 20% − 2% = 70 − 7 = 63.

Chains. a% of b% of c = $\frac{a \times b \times c}{10000}$. So 20% of 15% of 800 = $\frac{20 \times 15 \times 800}{10000}$ = 24.

What percent is it? "What percent of 250 is 45?" means $\frac{45}{250} \times 100$ = 18%.

A full question. Find 45% of 640 + 12.5% of 480 − 18% of 250.

• 45% of 640 = 288 (10% is 64, so 40% is 256, and 5% is 32)
• 12.5% of 480 = 480 ÷ 8 = 60
• 18% of 250 = 250% of 18 = 45
• Total: 288 + 60 − 45 = 303

Percentage increase, decrease and successive change belong to the Percentage chapter. Here a percentage is only a number to be worked out on the way to the answer.

When the percent is the unknown. In "?% of 840 = 126", divide first: 126 ÷ 840 = 0.15, so ? = 15.

What Comes in Place of the Question Mark?

This type comes up often in Tier 1. One part of an equation is replaced by "?", and you have to find it.

The method.

1. Work out the side without "?" completely.
2. Simplify whatever you can on the side with "?".
3. Undo the operations around "?" in reverse order. The last thing done to "?" is the first thing you undo.

Undoing steps. Find ? if (? ÷ 8) × 5 − 17 = 48.

• The last step was "− 17", so add 17: (? ÷ 8) × 5 = 65
• Undo "× 5": ? ÷ 8 = 13
• Undo "÷ 8": ? = 104

"?" in different places.

Use the options. You don't always need to solve fully. Two checks settle most of these fast.

• Unit digit. For 47 × 23 + ? = 1200, the product 47 × 23 ends in 1, so ? must end in 9. (47 × 23 = 1081 and ? = 119.)
• Size. For ? = 2856 ÷ 24 with options 109, 119, 129 and 139, note that 24 × 120 = 2880 is just above 2856. So ? is a little under 120, which means 119.

Plugging in. When "?" sits inside a bracket or under a power, putting each option into the question can be quicker than undoing the steps. Start with a middle-sized option, so one test tells you whether to go up or down.

"Exact value" questions need exact working. Rounding is allowed only in "approximate value" questions, where the numbers are ugly and the options are far apart; the identities and approximation section covers those.

Number-Pattern Sums: 7 + 77 + 777 + …

The column method. Write the numbers under one another and count how many times the digit appears in each column. For 7 + 77 + 777 + 7777:

• Ones column: four 7s = 28. Write 8, carry 2.
• Tens column: three 7s = 21, plus 2 = 23. Write 3, carry 2.
• Hundreds column: two 7s = 14, plus 2 = 16. Write 6, carry 1.
• Thousands column: one 7, plus 1 = 8.

The sum is 8638.

The 123… shortcut. 1 + 11 + 111 + 1111 = 1234, and 1 + 11 + 111 + 1111 + 11111 = 12345. So for any digit $a$, up to nine terms,

$$a + aa + aaa + \cdots \text{ (n terms)} = a \times 123\ldots n$$

That gives 7 + 77 + 777 + 7777 = 7 × 1234 = 8638, and 5 + 55 + 555 + 5555 + 55555 = 5 × 12345 = 61725. The shortcut breaks at ten terms, because the carries pile up: 1 + 11 + … (ten terms) = 1234567900, not 1234567890.

The general formula. [Tier 2] For $n$ terms,

$$a + aa + aaa + \cdots = \frac{a}{81}\left(10^{n+1} - 10 - 9n\right)$$

For 7 + 77 + 777 + 7777, $n = 4$, and $\frac{7}{81}(100000 - 10 - 36) = \frac{7}{81} \times 99954 = 8638$.

Decimal patterns. Line up the decimal places and add, or count the digits column by column again.

$$0.7 + 0.77 + 0.777 = 0.700 + 0.770 + 0.777 = 2.247$$

For $n$ terms the formula is

$$0.a + 0.aa + 0.aaa + \cdots = \frac{a}{81}\left(9n - 1 + 10^{-n}\right)$$

Tier 2 sometimes puts the answer in exactly this form. Check it on 0.7 + 0.77 + 0.777: $\frac{7}{81}(27 - 1 + 0.001) = \frac{7}{81} \times 26.001 = 2.247$.

Nines. Each 9-number is one less than a power of 10, so

$$9 + 99 + 999 + 9999 = (10 + 100 + 1000 + 10000) - 4 = 11110 - 4 = 11106$$

$$0.9 + 0.99 + 0.999 = 3 - (0.1 + 0.01 + 0.001) = 3 - 0.111 = 2.889$$

Patterns with subtraction. For 4444 − 444 + 44 − 4, pair the terms up: 4444 − 444 = 4000 and 44 − 4 = 40, so the answer is 4040.

Multiplying and Dividing Fast

Many simplification questions are built so that a number like 25, 125 or 99 makes the work short. Spot it and the long multiplication disappears.

The decimal forms work the same way: 2.5 = 10 ÷ 4 and 1.25 = 10 ÷ 8. So 2.5 × 36 = 360 ÷ 4 = 90, and 1.25 × 64 = 640 ÷ 8 = 80.

× 11 for a two-digit number. Write the first digit, then the sum of the two digits, then the last digit. For 63 × 11: 6, then 6 + 3 = 9, then 3, which gives 693. If the middle sum is 10 or more, carry the 1 into the first digit: 87 × 11 gives 8, 15, 7, so 957. For three digits, add neighbours: 352 × 11 is 3, 3 + 5, 5 + 2, 2, which gives 3872.

Take out the common factor. When the same number multiplies two or more terms, take it out first:

$$47 \times 68 + 47 \times 32 = 47 \times (68 + 32) = 4700$$

$$999 \times 999 + 999 = 999 \times (999 + 1) = 999000$$

$$12.5 \times 7.2 + 12.5 \times 2.8 = 12.5 \times 10 = 125$$

SSC often hides the common factor by moving it around. In 36 × 47 + 64 × 47 − 47 × 20, the 47 is in every term, so the value is 47 × (36 + 64 − 20) = 47 × 80 = 3760.

Products near a base like 98 × 102, and squares like $997^2$, are faster with identities, which are in the identities section.

Solved Examples

Question 1

Find the value of 64 ÷ 8 × 4 − 3 × 5 + 6. (a) −7 (b) 23 (c) 11 (d) −1

Answer. ÷ and × have equal rank, so I go left to right: 64 ÷ 8 = 8, then 8 × 4 = 32. Next, 3 × 5 = 15. Now 32 − 15 + 6, again left to right, gives 17 + 6 = 23. If I had done 8 × 4 first, I would get 64 ÷ 32 = 2 and land on −7, which is option (a). The answer is (b) 23.

Question 2

Simplify 72 ÷ 6 of 3 + 5 × 4 − 2.

Answer. "of" comes before ÷, so 6 of 3 = 18 first. Then 72 ÷ 18 = 4 and 5 × 4 = 20. Finally 4 + 20 − 2 = 22. Reading "of" as an ordinary × would give 72 ÷ 6 × 3 = 36 and a wrong answer of 54.

Question 3

Simplify $48 - [20 - \{16 - (9 - \overline{7 - 4})\}]$.

Answer. I start with the bar and work outwards. Bar: 7 − 4 = 3. Round bracket: 9 − 3 = 6. Curly bracket: 16 − 6 = 10. Square bracket: 20 − 10 = 10. Last step: 48 − 10 = 38.

Question 4

Find the value of −8 − (−5) × 3 + 12 ÷ (−4). (a) 4 (b) −20 (c) 10 (d) −26

Answer. The × and ÷ go first. (−5) × 3 = −15 and 12 ÷ (−4) = −3. So the line becomes −8 − (−15) + (−3) = −8 + 15 − 3 = 4. Treating −(−15) as −15 gives −26, which is option (d). The answer is (a) 4.

Question 5

Simplify $\frac{1}{2}$ of $\frac{3}{4}$ ÷ $\frac{3}{8}$ × $\frac{2}{3}$ + $\frac{1}{3}$. (a) $\frac{11}{6}$ (b) 1 (c) $\frac{2}{3}$ (d) $\frac{3}{2}$

Answer. "of" first: $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$. Then ÷ and × from left to right: $\frac{3}{8} \div \frac{3}{8} = 1$, and $1 \times \frac{2}{3} = \frac{2}{3}$. Adding $\frac{1}{3}$ gives 1. The answer is (b) 1. If I multiplied before dividing, I would get $\frac{3}{8} \div \frac{1}{4} = \frac{3}{2}$ and then $\frac{11}{6}$, which is option (a).

Question 6

Simplify $\frac{0.4 \times 0.4 + 0.02}{0.3 \times 0.3 + 0.03}$.

Answer. 0.4 × 0.4 = 0.16, so the top is 0.16 + 0.02 = 0.18. 0.3 × 0.3 = 0.09, so the bottom is 0.09 + 0.03 = 0.12. Multiplying top and bottom by 100 gives $\frac{18}{12} = \frac{3}{2}$, which is 1.5.

Question 7

Find the value of 0.0084 ÷ 0.12 + 2.5 × 0.04 × 0.3.

Answer. For the division I move both points two places: 0.84 ÷ 12 = 0.07. For the product, 25 × 4 × 3 = 300, and there are 1 + 2 + 1 = 4 decimal places, so it is 0.0300 = 0.03. The total is 0.07 + 0.03 = 0.1.

Question 8

Find 0.125 × 0.64 ÷ 0.025. (a) 0.32 (b) 3.2 (c) 32 (d) 0.032

Answer. 0.125 is $\frac{1}{8}$, so 0.125 × 0.64 = 0.64 ÷ 8 = 0.08. Then for 0.08 ÷ 0.025 I move both points three places, which gives 80 ÷ 25 = 3.2. The answer is (b) 3.2.

Question 9

If 73.4602 = 7 × A + 3 × B + 4 ÷ C + 6 × D + 2 ÷ E, find the value of A + B + C + D + E.

Answer. I match each digit to its place. The 7 is in the tens place, so 7 × A = 70 and A = 10. The 3 is in the ones place, so B = 1. The 4 is in the tenths place, worth 0.4, and 4 ÷ C = 0.4 gives C = 10. The 6 is in the hundredths place, worth 0.06, so 6 × D = 0.06 and D = 0.01. The thousandths digit is 0, so it has no term. The 2 is in the ten-thousandths place, worth 0.0002, so 2 ÷ E = 0.0002 and E = 10000. Adding them: 10 + 1 + 10 + 0.01 + 10000 = 10021.01.

Question 10

If $2.4a = 0.06b$, find $\frac{b - a}{b + a}$. (a) $\frac{39}{41}$ (b) $\frac{41}{39}$ (c) $\frac{1}{40}$ (d) $\frac{19}{21}$

Answer. $\frac{a}{b} = \frac{0.06}{2.4}$. Multiplying top and bottom by 100 gives $\frac{6}{240} = \frac{1}{40}$, so $b = 40a$. Then $\frac{b - a}{b + a} = \frac{39a}{41a}$. The answer is (a) $\frac{39}{41}$. If I flipped the ratio and took $a = 40b$, I would get a negative value, so it is worth checking which letter is bigger first.

Question 11

Find the value of 35% of 840 − 12.5% of 640 + 18% of 50.

Answer. 35% of 840: 10% is 84, so 30% is 252, and 5% is 42, which makes 294. 12.5% is $\frac{1}{8}$, and 640 ÷ 8 = 80. For 18% of 50 I swap it round to 50% of 18 = 9. So the value is 294 − 80 + 9 = 223.

Question 12

Find 20% of 35% of 1800.

Answer. 35% of 1800 = 630, and 20% of 630 is 630 ÷ 5 = 126. In one line it is $\frac{20 \times 35 \times 1800}{10000} = 126$.

Question 13

What comes in place of ? in (? ÷ 14) × 6 + 23 = 65?

Answer. I undo the steps starting from the last one. Take away 23: (? ÷ 14) × 6 = 42. Divide by 6: ? ÷ 14 = 7. Multiply by 14: ? = 98. Check: 98 ÷ 14 = 7, 7 × 6 = 42 and 42 + 23 = 65.

Question 14

45% of 380 + ? = 60% of 450 − 15. Find ?. (a) 99 (b) 69 (c) 84 (d) 114

Answer. Left side: 45% of 380 = 171 (10% is 38, so 40% is 152, and 5% is 19). Right side: 60% of 450 = 270, and 270 − 15 = 255. So 171 + ? = 255, which gives ? = 84. Forgetting the − 15 gives 99, which is option (a). The answer is (c) 84.

Question 15

$\frac{2}{5}$ of $\frac{3}{4}$ of ? = 216. Find ?.

Answer. $\frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10}$, so $\frac{3}{10}$ of ? = 216. Then ? = 216 × $\frac{10}{3}$ = 720.

Question 16

Find ? if $(?)^2 + 19 \times 4 = 15^2 - 5$.

Answer. Right side: $15^2 = 225$, and 225 − 5 = 220. Left side: 19 × 4 = 76. So $(?)^2 = 220 - 76 = 144$, and ? = 12. SSC options take the positive value.

Question 17

Find 6 + 66 + 666 + 6666 + 66666.

Answer. There are five terms, and 1 + 11 + 111 + 1111 + 11111 = 12345. So the sum is 6 × 12345 = 74070. In columns it comes out the same: the ones column has five 6s = 30 (write 0, carry 3), the tens 24 + 3 = 27 (write 7, carry 2), the hundreds 18 + 2 = 20 (write 0, carry 2), the thousands 12 + 2 = 14 (write 4, carry 1), and the ten-thousands 6 + 1 = 7.

Question 18

Find 0.4 + 0.44 + 0.444 + 0.4444.

Answer. I give every term four decimal places: 0.4000 + 0.4400 + 0.4440 + 0.4444. Reading the columns from the right, the ten-thousandths give 4, the thousandths 8, the hundredths 12 (write 2, carry 1), and the tenths 16 + 1 = 17 (write 7, carry 1 into the ones). The sum is 1.7284.

Question 19

Find the sum 3 + 33 + 333 + … up to 8 terms. (a) 37037037 (b) 3703701 (c) 37037034 (d) 37037340

Answer. Eight terms of 1 + 11 + 111 + … add up to 12345678. So the sum is 3 × 12345678 = 37037034. Stopping at seven terms gives option (b). The answer is (c) 37037034.

Question 20

Find 368 × 125 and 2736 ÷ 25 without long multiplication or division.

Answer. × 125 is the same as × 1000 then ÷ 8, so 368 × 125 = 368000 ÷ 8 = 46000. ÷ 25 is the same as × 4 then ÷ 100. 2736 × 4 = 10944, so 2736 ÷ 25 = 109.44.

Question 21

Find 4783 × 99 and 642 × 101.

Answer. × 99 is × 100 minus the number: 4783 × 99 = 478300 − 4783 = 473517. × 101 is × 100 plus the number: 642 × 101 = 64200 + 642 = 64842.

Question 22

Simplify 17.5 × 4.8 + 17.5 × 5.2 − 7.5 × 10. (a) 175 (b) 100 (c) 250 (d) 75

Answer. 17.5 is common to the first two terms, so they make 17.5 × (4.8 + 5.2) = 17.5 × 10 = 175. Then 175 − 75 = 100. The answer is (b) 100.