5.01a, 5.01c - Ratios and fractions

5.01a, 5.01c - Ratios and fractions

A drink contains two measures of concentrate for every three measures of water. What fraction is concentrate? Learn to simplify ratios in common units, write them as one to n and distinguish a comparison between parts from a fraction of the whole.

Ratios in the same units

A ratio records how quantities compare in a stated order: concentrate to water 2:32:3 means two equal-sized measures of concentrate for every three of water.

Equivalent ratios

A ratio compares quantities using a colon. Equivalent ratios multiply or divide every entry by the same nonzero number. Thus 2:32:3, 4:64:6 and 10:1510:15 describe the same comparison. For positive whole-number entries, divide by their highest common factor to get the simplest whole-number ratio.

For quantities of the same kind, convert to the same units before simplifying. 1 m:50 cm1\,\mathrm{m}:50\,\mathrm{cm} is 100:50=2:1100:50=2:1, not 1:501:50. The units then cancel because both values use the same unit. Keep the named order throughout.

Worked example 1: converting the larger unit

Question. Write 45 cm:1.2 m45\,\mathrm{cm}:1.2\,\mathrm{m} as a ratio in its simplest whole-number form.

Plan. Convert metres to centimetres, then divide both entries by their highest common factor.

1.2 m=120 cm45:120=(45÷15):(120÷15)=3:8\begin{aligned} 1.2\,\mathrm{m}&=120\,\mathrm{cm} \\ 45:120&=(45\div15):(120\div15) \\ &=3:8 \end{aligned}

Answer. 3:83:8.

Check. Multiplying both entries by 15 recovers 45:12045:120, the two lengths in centimetres. The first length is smaller, as 3<83<8.

Change only the decimal position in the second length. The unit conversion stays the same, but the relative sizes reverse.

Worked example 2: the first quantity is now larger

Question. Write 45 cm:0.12 m45\,\mathrm{cm}:0.12\,\mathrm{m} in its simplest whole-number form.

Plan. Convert the second length before comparing the numbers.

0.12 m=12 cm45:12=(45÷3):(12÷3)=15:4\begin{aligned} 0.12\,\mathrm{m}&=12\,\mathrm{cm} \\ 45:12&=(45\div3):(12\div3) \\ &=15:4 \end{aligned}

Answer. 15:415:4.

Check. 15÷4=3.7515\div4=3.75, and 45÷12=3.7545\div12=3.75. Both comparisons say the first length is 3.75 times the second.

Decimals and fractions are allowed in a ratio. Clear them by multiplying every entry by a suitable number: 0.6:1.5=6:15=2:50.6:1.5=6:15=2:5, and 12:34=2:3\frac12:\frac34=2:3 after multiplying by 4. Adding the same number does not preserve a ratio: 2:32:3 and 3:43:4 are different comparisons.

A zero entry can be meaningful: no red counters and five blue gives 0:5=0:10:5=0:1. The pair 0:00:0 gives no fixed relative composition. Ordinary sharing problems here use positive entries and positive amounts.

Ratios in the form one to n

A ratio with first entry 1 tells you how much of the second quantity goes with one unit of the first.

To turn a:ba:b into 1:n1:n, divide both entries by aa. The second entry becomes b÷ab\div a, which may be a fraction or decimal. This form is different from the simplest whole-number form, so read which form is requested.

One to n

For positive aa,

a:b=1:ba.a:b=1:\frac ba.

The first entry must be nonzero to divide by it. The second entry need not exceed 1 or be an integer. Not on the 2027 formula sheet: use equivalent-ratio reasoning.

A ratio is ordered. Reversing the quantities changes the unit comparison.

Worked example 3: a second entry above one

Question. Write 12:3012:30 in the form 1:n1:n.

Plan. Divide both entries by the first entry, 12.

12:30=(12÷12):(30÷12)=1:2.5\begin{aligned} 12:30&=(12\div12):(30\div12) \\ &=1:2.5 \end{aligned}

Answer. 1:2.51:2.5, so n=2.5n=2.5.

Check. Multiplying both entries by 12 gives 12:3012:30 again.

Now reverse the same comparison. The unit ratio becomes the reciprocal of the previous second entry.

Worked example 4: a second entry below one

Question. Write 30:1230:12 in the form 1:n1:n.

Plan. The first entry is now 30, so divide both entries by 30.

30:12=(30÷30):(12÷30)=1:0.4\begin{aligned} 30:12&=(30\div30):(12\div30) \\ &=1:0.4 \end{aligned}

Answer. 1:0.41:0.4, so n=0.4n=0.4.

Check. Multiplying by 30 gives 30:1230:12. Also, 2.5×0.4=12.5\times0.4=1, as expected from reversing the comparison.

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