Simplify: 0.52 x 0.24 x 0.5

0.0624. Let the three factors be 0.52, 0.24, and 0.5. First multiply them as if there were no decimal points at all. Then count the total decimal places across all three factors: 5 place(s) in total. Place the decimal point that many digits from the right in the whole-number product. This gives 0.52 x 0.24 x 0.5 = 0.0624. Check: multiplying the three decimals directly also gives 0.0624.

What is 851.64 divided by 47, rounded to two decimal places?

18.12. Let the dividend be 851.64 and the divisor be 47. We need 851.64 / 47. Performing the division and rounding to two decimal places gives 18.12. Check: 47 x 18.12 = 851.64, close to 851.64, confirming the quotient.

What is 66.93 divided by 100?

0.6693. Let x = 66.93. We need x divided by 100. 100 is 10^2, since it has 2 zero(s). Divided by a power of 10 shifts the decimal point 2 place(s) to the left. Shifting x = 66.93 this way gives 0.6693. Check: 0.6693 multiplied by 100 should give back 66.93.

A fraction whose denominator is a power of 10 is called what?

A decimal fraction. A decimal fraction is defined as a fraction whose denominator is a power of 10. Examples include tenths (1/10), hundredths (1/100), and thousandths (1/1000). This differs from a vulgar fraction, whose denominator can be any whole number. So the correct term here is a decimal fraction.

A recurring decimal in which ALL the figures after the decimal point repeat is called what?

A pure recurring decimal. A pure recurring decimal is one where every digit after the point belongs to the repeating block. For example, 0.777... repeats from the very first digit. This is different from a mixed recurring decimal, which has non-repeating digits first. So the correct term here is a pure recurring decimal.

To convert a pure recurring decimal into a vulgar fraction, the denominator uses how many nines?

As many nines as there are repeating figures. The rule: numerator = the repeating block of digits, denominator = as many 9's as there are repeating digits. For example, 0.555... has one repeating digit, so it equals 5/9. For example, 0.232323... has two repeating digits, so it equals 23/99. So the denominator needs as many nines as there are repeating figures.

Decimal Fractions: Formulas, Shortcuts and Solved Examples

Decimal fractions appear in almost every aptitude paper. The ideas are small, but slips with the point cost marks.

This lesson gives each rule with the reason it works. It then solves four examples step by step and lists the traps that cost marks.

The idea in brief

Shift: one place for each zero in 10, 100, 1000.
Multiply: add the decimal places.
Divide: keep the point straight above.
Compare: turn fractions into decimals.
Recurring: nines for the repeating digits.

What Decimals and Fractions Are

What is a decimal fraction?

A decimal fraction is a fraction whose bottom number is 10, 100, 1000 and so on. We write it with a point. So 25/100 is 0.25.

Each place after the point is ten times smaller than the one before it.

Why does the point move when you multiply by 10?

Every place value is ten times the place to its right. Multiplying by 10 pushes each digit one place up. That looks like the point moving right.

Dividing by 10 does the reverse. Each digit drops one place.

37.25 divided by 10 gives 3.725. The digits stay in order. The decimal point moves one place to the left.

Check 1

Dividing 37.25 by 10 gives 372.5.

Core Rules and Why They Work

How do you multiply or divide by 10, 100 or 1000?

Move the point right to multiply and left to divide. Move it one place for each zero.

Why it works: each zero is one factor of 10, and each factor of 10 shifts every digit by one place.

How do you multiply decimals?

Ignore the points, multiply the whole numbers, then put the point back. Count the decimal places in both numbers and add them.

Why it works: 0.52 is 52/100 and 0.24 is 24/100. The bottoms multiply to 10000, which is four decimal places.

How do you divide a decimal by a whole number?

Divide as usual and put the point in the answer straight above the point in the number.

To round to two places, work to three places and look at the third digit.

Why it works: the place values line up the same way as in whole-number division.

How do you compare fractions?

Turn each fraction into a decimal and compare the decimals.

Why it works: decimals share one scale, so the larger number is clear at once. Fractions with different bottoms are hard to judge by eye.

Four fractions on a number line from 0 to 1. The smallest is 3/9 and the biggest is 5/6. Their difference is 1/2.

How do you turn a recurring decimal into a fraction?

Pure recurring: write the repeating digits over as many nines as there are repeating digits. So 0.3 repeating is 3/9, and 0.45 repeating is 45/99.Mixed recurring: (all digits − the non-repeating digits), over nines then zeros. So 0.1666… is (16 − 1)/90 = 1/6.

Why it works: multiply by 10 to the power of the repeat length. Subtract the original. The repeating tail cancels.

Check 2

A pure recurring decimal has some digits that do not repeat.

Four Solved Examples

Easy, step by step: how many times is 3 greater than 0.3?

Step 1: divide the big number by the small one. 3 ÷ 0.3.

Step 2: multiply both by 10 to get 30 ÷ 3.

Step 3: the answer is 10 times.

Medium, step by step: simplify 0.52 × 0.24 × 0.5.

Step 1: 52 × 24 = 1248. The two numbers have 2 + 2 = 4 decimal places. So 0.52 × 0.24 = 0.1248.

Step 2: 1248 × 5 = 6240. Together 0.1248 and 0.5 have 4 + 1 = 5 decimal places.

Step 3: so 0.1248 × 0.5 = 0.0624.

Exam level, step by step: what is 851.64 ÷ 47, rounded to two decimal places?

Step 1: 47 goes into 851 eighteen times, because 47 × 18 = 846. The remainder is 5.

Step 2: bring down the 6 and 4. The rest is 5.64, and 5.64 ÷ 47 = 0.12.

Step 3: the quotient is 18.12, so the answer is 18.12.

Exam level, step by step: find the difference between the biggest and the smallest of 5/10, 4/6, 3/9 and 5/6.

Step 1: as decimals they are 0.5, 0.667, 0.333 and 0.833.

Step 2: the biggest is 5/6 and the smallest is 3/9.

Step 3: 5/6 − 3/9 = 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2.

Check 3

To find how many times a is greater than b, divide a by b.

Common Traps and Shortcuts

Which traps cost marks?

Moving the point the wrong way. Multiplying makes a number bigger, so the point moves right.

Losing a zero. 0.0624 and 0.624 look close but are ten times apart.

Comparing fractions by the top number alone. 5/10 is smaller than 4/6.

Using the wrong number of nines. Count only the repeating digits.

Which shortcuts save time?

Count the decimal places first. Then multiply the whole numbers.

Round only at the last step.

For “how many times”, the answer is a power of 10 when the digits are the same.

Turn fractions to decimals only when the bottoms differ.

What patterns come up in exams?

Five patterns repeat. Shifting the point, multiplying decimals, dividing and rounding, comparing fractions, and recurring decimals.

Check the answer size. Estimate first, then match your answer to it.

Write the number of decimal places in the margin before you start.

Check 4

The product of two numbers with 2 and 3 decimal places has 5 decimal places.

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