Sharing in a ratio questions with answers

Questions

1. Question 1Share a total in a ratio

   Kumar and Siti share 165 stamps in the ratio $7 : 4$. Find the number of stamps Siti gets.

   Show the worked solution

   1. Step 1: Count the total number of units$\text{Total} = 7 + 4$$\phantom{\text{Total}} = 11 \text{ units}$
   2. Step 2: Find 1 unit$11 \text{ units} \rightarrow 165$$1 \text{ unit} \rightarrow \dfrac{165}{11} = 15$
   3. Step 3: Multiply by Siti's number of units$4 \text{ units} \rightarrow 4 \times 15 = 60$Siti gets $60$ stamps.
2. Question 2One share is known

   The ratio of Devi's savings to Kumar's savings is $1 : 3$. Devi has $15. Find the total amount of money they have.

   Show the worked solution

   1. Step 1: Match the known amount to its number of units$1 \text{ unit} \rightarrow 15$
   2. Step 2: Find 1 unit (in dollars)$1 \text{ unit} \rightarrow \dfrac{15}{1} = 15$
   3. Step 3: Multiply by the total number of units$4 \text{ units} \rightarrow 4 \times 15 = 60$Altogether they have $60.
3. Question 3The difference is known

   The ratio of Ali's stamps to Farah's stamps is $3 : 4$. Farah has 4 more stamps than Ali. Find the number of stamps Farah has.

   Show the worked solution

   1. Step 1: The difference in units$4 - 3 = 1 \text{ unit}$
   2. Step 2: Find 1 unit$1 \text{ unit} \rightarrow 4$$1 \text{ unit} \rightarrow \dfrac{4}{1} = 4$
   3. Step 3: Multiply by Farah's number of units$4 \text{ units} \rightarrow 4 \times 4 = 16$Farah has $16$ stamps.
4. Question 4Share in a three-part ratio

   Siti, Devi and Kumar share $120 in the ratio $1 : 2 : 3$. Find the amount Siti receives.

   Show the worked solution

   1. Step 1: Count the total number of units$\text{Total} = 1 + 2 + 3$$\phantom{\text{Total}} = 6 \text{ units}$
   2. Step 2: Find 1 unit (in dollars)$6 \text{ units} \rightarrow 120$$1 \text{ unit} \rightarrow \dfrac{120}{6} = 20$
   3. Step 3: Multiply by Siti's number of units$1 \text{ unit} \rightarrow 1 \times 20 = 20$Siti receives $20.
5. Question 5Combine a : b and b : c

   The ratio of Mei Ling's marbles to Ali's marbles is $3 : 4$. The ratio of Ali's marbles to Farah's marbles is $3 : 7$. Find the ratio of Mei Ling's marbles to Ali's marbles to Farah's marbles.

   Show the worked solution

   1. Step 1: Make Ali's number of units the same in both ratios: use the LCM$\text{LCM of } 4 \text{ and } 3 = 12$
   2. Step 2: Scale each ratio so that the middle term matches$\text{Mei Ling} : \text{Ali} = 3 : 4$$\phantom{\text{Mei Ling} : \text{Ali}} = 3 \times 3 : 4 \times 3$$\phantom{\text{Mei Ling} : \text{Ali}} = 9 : 12$$\text{Ali} : \text{Farah} = 3 : 7$$\phantom{\text{Ali} : \text{Farah}} = 3 \times 4 : 7 \times 4$$\phantom{\text{Ali} : \text{Farah}} = 12 : 28$
   3. Step 3: Put the three together$\text{Mei Ling} : \text{Ali} : \text{Farah} = 9 : 12 : 28$
6. Question 6Before and after: the ratio changes

   The ratio of the number of boys to the number of girls is $5 : 2$. After 18 more girls join the club, the ratio becomes $5 : 3$. Find the number of girls now.

   Show the worked solution

   1. Step 1: The number of boys does not change: make its units the same in both ratios (LCM)$\text{LCM of } 5 \text{ and } 5 = 5$
   2. Step 2: Scale both ratios$\text{Before: } 5 : 2 = 5 : 2$$\text{After: } 5 : 3 = 5 : 3$
   3. Step 3: The 18 that joined are the extra units of girls$3 - 2 = 1 \text{ units} \rightarrow 18$$1 \text{ unit} \rightarrow \dfrac{18}{1} = 18$
   4. Step 4: Now: girls' units after × 1 unit$3 \times 18 = 54$There are now $54$ girls.
7. Question 7A fraction of the whole, then a ratio of the rest

   Of the fruits in a basket, $\dfrac{1}{6}$ are mangoes. The rest are apples and oranges in the ratio $1 : 2$. There are 20 oranges. Find the number of mangoes.

   Show the worked solution

   1. Step 1: The rest is what is left of the whole: 1 − 1/6$1 - \dfrac{1}{6} = \dfrac{5}{6}$
   2. Step 2: The oranges are their share of the units of the rest: multiply the two fractions$\text{oranges} = \dfrac{2}{3} \times \dfrac{5}{6}$$\phantom{\text{oranges}} = \dfrac{10}{18}$ of the whole
   3. Step 3: The known number is that fraction of the whole: divide by the fraction (turn it upside down and multiply)$\text{Whole} = 20 \div \dfrac{10}{18}$$\phantom{\text{Whole}} = 20 \times \dfrac{18}{10}$$\phantom{\text{Whole}} = 36$
   4. Step 4: The mangoes are 1/6 of the whole$\dfrac{1}{6} \times 36 = 6$There are $6$ mangoes.
8. Question 8Combine two ratios, then one more step

   The ratio of the number of stamps Chen Wei has to the number Jun Jie has is $2 : 3$. The ratio of the number of stamps Jun Jie has to the number Ravi has is $4 : 5$. Chen Wei then gives $\dfrac{1}{4}$ of his stamps to Ravi, and Ravi now has 68 stamps. Find the total number of stamps the three of them have.

   Show the worked solution

   1. Step 1: Make Jun Jie's units the same in both ratios (LCM), then join them$\text{LCM of } 3 \text{ and } 4 = 12$$\text{Chen Wei} : \text{Jun Jie} : \text{Ravi} = 8 : 12 : 15$
   2. Step 2: Chen Wei gives away 1/4 of his units; Ravi gains them$\dfrac{1}{4} \times 8 = 2 \text{ units}$$Ravi: 15 + 2 = 17 \text{ units} \rightarrow 68$
   3. Step 3: Find 1 unit; the total does not change when stamps are passed around$1 \text{ unit} \rightarrow \dfrac{68}{17} = 4$$\text{Total} = (8 + 12 + 15) \times 4$$\phantom{\text{Total}} = 140$They have $140$ stamps altogether.
9. Question 9A : (B + C), and the difference between B and C

   Siti, Devi and Kumar share $240. The ratio of Siti's share to the total of Devi's and Kumar's shares is $2 : 3$. Devi receives $20 less than Kumar. Find the amount Devi receives.

   Show the worked solution

   1. Step 1: Devi and Kumar together have 3 of the 5 units (in dollars)$\dfrac{3}{5} \times 240 = 144$
   2. Step 2: Take the difference away, and the rest splits equally: that gives Devi's share$Devi = \dfrac{144 - 20}{2}$$\phantom{Devi} = 62$Devi receives $62.
10. Question 10An unknown term, and a share as a percentage

    Siti, Devi and Kumar share some money in the ratio $x : 4 : 5$. Devi receives $120, and Siti's share is 50% of the total. Find the total amount of money shared.

    Show the worked solution

    1. Step 1: Devi and the third person together have the other 50% of the total$100\% - 50\% = 50\%$
    2. Step 2: Those 9 units are 50% of the total; so 50% of the total is how many units?$x = \dfrac{50}{50} \times 9$$\phantom{x} = 9$
    3. Step 3: 1 unit from Devi's share, then the total (in dollars)$1 \text{ unit} = \dfrac{120}{4}$$\phantom{1 \text{ unit}} = 30$$\text{Total} = (9 + 4 + 5) \times 30$$\phantom{\text{Total}} = 540$The total amount shared is $540.
11. Question 11Sharing in a ratio with letters

    Two sisters share $\$n$ in the ratio $p : q$, where $p > q$. Find an expression, in terms of $n$, $p$ and $q$, for the smaller share, in dollars.

    Show the worked solution

    1. Step 1: Count the units, just as with numbers: x + y units in all$\text{Total} = p + q \text{ units}$
    2. Step 2: One unit is the whole divided by the number of units$1 \text{ unit} = \dfrac{n}{p + q}$
    3. Step 3: The smaller part is the q units$q \text{ units} \rightarrow q \times \dfrac{n}{p + q} = \dfrac{nq}{p + q}$
    4. Step 4: Multiply the units wanted by 1 unit$= \dfrac{nq}{p + q}$