FN11, FR4–FR5, FR7, HN11, HR4–HR5, HR7 - Writing and simplifying ratios

FN11, FR4–FR5, FR7, HN11, HR4–HR5, HR7 - Writing and simplifying ratios

Learn what a ratio compares, how to write equivalent ratios, and how a ratio connects to fractions. FR5 and HR5 cover ratio notation and simplification; FN11, FR4, FR7 and their Higher counterparts connect the same quantities through fractions and multiplication.

Comparing quantities

A bag contains red and blue counters in the ratio 2:32:3. Does that mean 23\frac23 of the bag is red? First decide what the two numbers compare.

Ratio

A ratio compares quantities in a stated order. Red to blue in the ratio 2:32:3 means two equal-sized parts of red for every three of blue. It does not specify the actual number of counters.

The bag might contain 22 red and 33 blue counters, or 2020 red and 3030 blue counters. The comparison is unchanged. Reversing the order reverses the entries: blue to red is 3:23:2.

A part-to-part ratio compares one group with another. A part-to-whole ratio compares a group with the total, so add all the groups before writing the whole.

Worked example 1: part to part

Question. There are 66 red counters and 99 blue counters. Write the ratio of red counters to blue counters.

Plan. Keep the order “red, then blue” when recording the counts.

red:blue=6:9\text{red}:\text{blue}=6:9

Answer. 6:96:9.

Check. The second entry is larger, matching the larger blue group. The reversed ratio 9:69:6 would answer blue to red.

Keep the same counters but change the second quantity to the whole bag.

Worked example 2: part to whole

Question. There are 66 red counters and 99 blue counters. Write the ratio of red counters to all the counters.

Plan. Count the whole bag before writing the second entry.

total=6+9=15red:all=6:15\begin{aligned} \text{total}&=6+9=15\\ \text{red}:\text{all}&=6:15 \end{aligned}

Answer. 6:156:15.

Check. There are 99 non-red counters, so the whole must exceed the red group by 99, as 15−6=915-6=9.

When comparing measurements of the same kind, use matching units. For example, the ratio of 1.5 m1.5\,\mathrm m to 30 cm30\,\mathrm{cm} is 150:30150:30, after changing metres to centimetres. Writing 1.5:301.5:30 would compare numbers attached to different-sized units.

Watch out

In a red-to-blue ratio 2:32:3, the total is 55 parts, not 33. Label what each entry means before using it.

Write both comparisons for a fresh bag.

Simplifying ratios

Scaling every entry of a ratio by the same positive factor keeps the comparison unchanged. It is like resizing a recipe: every ingredient changes together.

Equivalent and simplest ratios

Equivalent ratios describe the same relative quantities, such as 6:96:9 and 2:32:3. A ratio in simplest whole-number form has whole-number entries with no common factor greater than 11.

Divide by the highest common factor to reach simplest form in one step. For three entries, divide all three: 12:18:30=2:3:512:18:30=2:3:5, using a common factor of 66.

Worked example 3: a common factor

Question. Simplify 18:2418:24.

Plan. Divide both entries by their highest common factor, 66.

18:24=(18÷6):(24÷6)=3:418:24=(18\div6):(24\div6)=3:4

Answer. 3:43:4.

Check. Multiplying both entries by 66 recovers 18:2418:24; 33 and 44 have no common factor greater than 11.

Now change the entries to decimals. First make them whole numbers by the same multiplication.

Worked example 4: decimal entries

Question. Simplify 0.18:0.240.18:0.24.

Plan. Multiply both entries by 100100, then simplify the resulting whole-number ratio.

0.18:0.24=18:24scale both entries by 100=3:4divide both by 6\begin{aligned} 0.18:0.24&=18:24 &&\text{scale both entries by }100\\ &=3:4 &&\text{divide both by }6 \end{aligned}

Answer. 3:43:4.

Check. Both 0.18÷30.18\div3 and 0.24÷40.24\div4 equal 0.060.06, so the parts have equal value.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account