Lesson Guide
Explanation
Test writers choose numbers that hide easy factors. Thirty-six times twenty-five looks like long multiplication, but thirty-six is nine times four, and four times twenty-five is one hundred, so the answer is nine hundred. For forty-two times thirty-five, divided by fourteen times fifteen, do not multiply anything out. Forty-two divided by fourteen is three. That leaves three times thirty-five, which is one hundred and five, divided by fifteen, which is seven. Cancel first and the numbers stay small.
Your Turn
Work out 16 × 75 and (27 × 32) ÷ (36 × 8). Check: 4 × 4 × 25 × 3 = 1200; 27 ÷ 36 = 3/4 and 32 ÷ 8 = 4, so the answer is 3.
Watch Out
Multiplying out first: 42 × 35 = 1470 and 14 × 15 = 210, and then you still have to divide 1470 by 210. Cancel first.
Explanation
When two numbers sit the same distance either side of a round number, multiply them with the difference of two squares. Forty-seven and fifty-three are both three away from fifty. So forty-seven times fifty-three is fifty squared minus three squared, which is two thousand five hundred minus nine, or two thousand four hundred and ninety-one. Squaring a number near a round one is different. Fifty-nine squared is sixty minus one, all squared. That is three thousand six hundred, minus two times sixty, plus one, which gives three thousand four hundred and eighty-one.
Your Turn
Work out 68 × 72 and 312. Check: 702 − 22 = 4900 − 4 = 4896; (30 + 1)2 = 900 + 60 + 1 = 961.
Watch Out
(60 − 1)2 is not 602 − 12. That loses the middle term: 592 = 3481, not 3599.
Explanation
On a no-calculator paper, a percentage is almost always a friendly fraction in disguise. Twelve and a half percent is one eighth, so thirty-seven and a half percent is three eighths. Three eighths of sixty-four is sixty-four divided by eight, which is eight, times three, which is twenty-four. Here is a trick: x percent of y equals y percent of x. Eighteen percent of fifty is the same as fifty percent of eighteen, which is nine. And a percentage change is a multiplier. A forty percent rise multiplies by seven fifths. A fifteen percent fall multiplies by seventeen twentieths.
Your Turn
Work out 87.5% of 48 and 14% of 50. Check: 48 ÷ 8 × 7 = 42; 50% of 14 = 7.
Watch Out
A 40% rise then a 40% fall does not return to the start: 7/5 × 3/5 = 21/25, an overall fall of 16%.
Explanation
When the options are far apart, you do not need the exact answer. You only need its size. Round every number to one significant figure. Six point oh two is about six, thirty-nine point eight is about forty, and point three stays point three. Six times forty is two hundred and forty. Dividing by point three is the same as multiplying by ten and dividing by three, so you get two thousand four hundred divided by three, which is eight hundred. Only one option is anywhere near that, so you are done.
Your Turn
Which is closest to 0.81 × 0.0398? A 0.0032 B 0.032 C 0.32 D 3.2. Check: 0.8 × 0.04 = 0.032, so B.
Watch Out
Dividing by a number less than 1 makes the answer bigger. 240 ÷ 0.3 is 800, not 80 and not 72.
Explanation
A multiple-choice question hands you the answer. It is one of the four options, so you can check them instead of solving. For x times x plus five equals eighty-four, try six: six times eleven is sixty-six, too small. Try seven: seven times twelve is eighty-four. Done. Last digits are even faster. Twenty-three times seventeen times thirteen must end in the same digit as three times seven times three, which is sixty-three, so it ends in three. Any option that does not end in three is out. Odd times odd is odd, and a perfect square can only end in zero, one, four, five, six or nine. Quiz and exercises now.
Your Turn
Which of 2701, 2704, 2706, 2708 could be 522? Check: 522 ends in 4 (2 × 2), and 502 = 2500, so 2704.
Watch Out
Substituting checks an option; it does not find every solution. If a question asks for all solutions, test every option or solve properly.
📌 Key Points — this lesson in a nutshell
- Factor first, multiply last. Split numbers into friendly factors (4 × 25 = 100, 8 × 125 = 1000) and cancel before you compute.
- Difference of two squares: (a − b)(a + b) = a2 − b2, so 47 × 53 = 502 − 32 = 2491. But (a − b)2 keeps its middle term.
- Percentages as fractions: 12.5% = \(\dfrac{1}{8}\), 37.5% = \(\dfrac{3}{8}\), 62.5% = \(\dfrac{5}{8}\), 20% = \(\dfrac{1}{5}\). And x% of y = y% of x.
- Estimate before you calculate: round each number to one significant figure and cross out every option of the wrong size.
- Work back from the options: check the last digit, odd or even, and the size, then substitute the survivors.
Step 1: Introduction
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