Free How to Pass Numerical Reasoning by Heidi Smith Page A

1.
    Using ratios to find actual quantities
    A frequently tested concept is the use of ratios to find actual quantities. Ratios only tell you the relationship between numbers, not actual quantities. However, if you know the actual quantity of either a part or the whole, you can determine the actual quantity of the other parts.
    Worked example
    In a bag of 60 green and red jellybeans, the ratio of red jellybeans to green jellybeans is 2 : 3. How many are red?
    Method 1
    The parent set (the whole) is ‘all jellybeans in the bag’. The subsets (the parts) are ‘red jellybeans’ and ‘green jellybeans’. You know the actual value of the whole and the ratio between the parts, so you can work out the actual values of each part:
Part (red)
+
Part (green)
=
Whole
2
+
3
=
5
    This tells you that for every 5 parts, 2 are red and 3 are green. To find the actual quantity of red jellybeans, multiply the ratio of red jellybeans by the actual value of the whole.

    Method 2
    You can solve this with simple algebra.
    One part = x . Therefore 2 x (red) + 3 x (green) = 60 (all jellybeans).
    5 x = 60
    x = 60 ÷ 5 = 12
    2 x = 24 red jellybeans.
    Worked example
    A certain spice mix contains 66 g of a mix of cumin and coriander in the ratio of 4 : x . If 1 part cumin = 6 g, what is the ratio of cumin to coriander?
    The parent set is ‘66 g of spice mix’ and consists of two subsets, cumin and coriander.
    1 part cumin = 6 g, so 4 × 6 g cumin = 4 parts (24 g). If 24 g of the mix is cumin, then 66 g – 24 g (42 g) is coriander.
    The ratio of actual quantities of cumin to coriander = 24 g : 42 g. Divide both sides by 6 to comply with the format of the ratio in the question: 24 : 42 = 4 : 7. The ratio of cumin to coriander is 4 : 7.
    Worked example
    At the Tedbury Rolling-Rollers, a race open to skaters and bladers, the ratio of skaters to all racers is 2 to 3. There are a total of 240 Rolling-Rollers in the race. How many are on skates?
    You are given a ‘part to whole’ ratio and asked to work out the actual value of a part:

    The total number of competitors is 240. Multiply the ratio (expressed as a fraction) by the actual value of the whole number to find the number of skaters:
    There are 160 skaters in the race.
    Worked example
    After a hot whites wash, the ratio of the pink socks to all socks emerging from the washing machine is 3 : 5. 15 pairs of socks were put in the washing machine at the start of the programme. How many socks are not pink at the end of the wash? (Assume that all the socks that go into the washing machine also come out of it.) 15 pairs of socks went into the washing machine = 30 socks. The question tells you that the ratio of pink socks to all socks = 3 : 5. Express the ratio as a fraction:
    If 3 ⁄ 5 of the socks are pink, then 2 ⁄ 5 of the socks are not pink. Multiply 2 ⁄ 5 by the actual value of the whole:
    Number of non-pink socks = 12 (or 6 pairs).
    Ratios practice questions
    Give all your answers in the ratio format x : y.
    Q1 In a school there is one qualified teacher for every 32 students. What is the ratio of students to qualified teachers?
    Q2 Last season, Southwold United won 18 games and lost 9 games. What is the ratio of games lost to games won?
    Q3 It takes 1 hour 20 minutes to bake a potato and 45 minutes to bake a pie. What is the ratio of time taken to bake a potato to time taken to bake a pie?
    Q4 Dave’s training schedule requires him to increase his weekly mileage by 50 miles every week. In Week 2 he cycles 250 miles. What is the ratio of miles cycled in week 2 to miles cycled in week 3?
    Q5 What is the ratio of currants to sultanas in a fruitcake consisting of 1 lb 2 oz currants and 12 oz sultanas? (There are 16 oz in 1 lb.) Q6 In a musical ensemble, the ratio of stringed instruments to all other instruments is 8 : 3. What is the ratio of stringed instruments to all instruments in the ensemble?
    Q7 At a rock concert, 250 out of the capacity crowd of 30,000 are left-handed. What fraction of the