Another Peek Beyond the Point: questions with answers

Divide distance by time: 3600 ÷ 15 = 240 metres per minute. Then divide by 1000 to convert metres to kilometres, giving 0.24 kilometres per minute.

The total gain is 3.2 × 375 = 1200 seconds. Divide by 60 seconds per minute to get 20 minutes. A small daily gain accumulates; it is not zero.

The smaller factor is 0.24. Multiplying it by 0.5, between 0 and 1, takes only a proper part of it. Thus the product 0.12 is less than 0.24 and therefore less than both positive factors.

The multiplier 9.5 is greater than 1, so the product is greater than 1.5. A decimal point alone does not determine whether a product increases. Multiplying a positive number by less than 1 makes it smaller; multiplying by 1 leaves it unchanged.

10 has one zero, so move the decimal point one place to the left. Write zeroes as needed to hold empty places, giving 12.3. Count only the zeroes in 10, not all its digits. Since 123 is positive and 10 is greater than 1, this quotient is smaller than 123.

The divisor 100 has two zeroes, so move the decimal point two places left. Each one-place move divides the value by 10. Moving right would multiply instead.

In 0.500, each original digit has the same place value as in 0.5. The added zeros represent zero parts, so 0.5 = 0.500. Inserting zeros elsewhere can change the place values of other digits.

The denominator 1000 requires 3 decimal places. Place the numerator digits so the last digit occupies that final place, adding leading zeros as needed. This gives 0.067. Count zeros in the denominator, not all its digits.

Whole-number division gives 331 with remainder 1. Regroup the remaining 1 ones as 10 tenths and continue dividing by 4 until the remainder is zero. The exact decimal quotient is 331.25. The remainder counts ones, so writing it straight after the decimal point gives the wrong value. An answer with a remainder is not the requested decimal form.

Divide 415 metres by 4 to obtain 103.75 metres each. The remaining 3 metres must also be shared. Writing 3 directly after the decimal point confuses a remainder with decimal digits.

Multiply 18 by 12 to obtain 216. The two factors require seven places after the decimal point in total. Add leading zeros where needed to obtain 0.0000216. Omitting these zeros changes the place values.

Add the decimal-place counts of the two factors before removing any trailing zeros from the result. Here two places and one place add to three places. The fraction denominators 100 and 10 multiply to 1000.

Since 0.01 × 100 = 1, dividing by 0.01 is multiplying by 100. Thus 25 ÷ 0.01 = 25 × 100 = 2500. The missing multiplier is 100.

The fraction that never ends is 1/3. In the long division for 1/3 the remainder comes back instead of reaching 0, so the answer runs on with the digit 3 repeating. The other three divisions each reach a remainder of 0 and stop: 1/8 = 0.125, 1/20 = 0.05 and 1/125 = 0.008. A big denominator does not make a decimal run on.

Dividing 128 by 0.4 counts how many 0.4 fit into 128. Since 0.4 is less than 1, each piece is smaller than one whole unit, so more pieces fit than there are units. That is why the quotient 320 is larger than 128. The decimal point by itself decides nothing: 2.5 also has one, and dividing by 2.5 makes the answer smaller. The dividend does not decide either, since every positive dividend grows when divided by 0.4.