Ratio Problems: Additional Practice (Set 1)

The shared term $B$ is $6$ in the first ratio and $8$ in the second, so scale each to the least common multiple $24$: $5 : 6 = 20 : 24$ and $8 : 9 = 24 : 27$.

$A : B : C = 20 : 24 : 27$

Write red as $4k$ and blue as $7k$. The red count is unchanged, so $4k : (7k + 6)$ equals $4 : 9$, and cross-multiplying gives the equation in $k$.

$9(4k) = 4(7k + 6) \implies 36k = 28k + 24 \implies k = 3$

Red is $4k = 12$.

Scale $Q$ to the least common multiple of $4$ and $8$, which is $8$: $3 : 4 = 6 : 8$, so $P : Q : R = 6 : 8 : 5$. The parts total $19$, so $19k = 9500$ and $k = 500$.

$19k = 9500 \implies k = 500, \quad 5k = 2500$

So $R$ gets $2500$ dollars.

Write the numbers as $9k$ and $5k$. After subtracting $8$ from each, $(9k - 8) : (5k - 8)$ equals $7 : 3$, so cross-multiply.

$3(9k - 8) = 7(5k - 8) \implies 27k - 24 = 35k - 56 \implies k = 4$

The smaller original number is $5k = 20$.

The parts total $4 + 5 + 6 = 15$, so $15k = 180$ and $k = 12$. The smallest angle is the $4$ term.

$15k = 180 \implies k = 12, \quad 4k = 48$

The smallest angle is $48$ degrees.

The parts total $1 + 2 + 9 = 12$, so $12k = 60$ and $k = 5$. Sand is the $9$ term.

$12k = 60 \implies k = 5, \quad 9k = 45$

You need $45$ kg of sand.

Write Maria's savings as $7k$ and Nadia's as $4k$. After Maria spends $60$, they are equal, so $7k - 60 = 4k$.

$7k - 60 = 4k \implies 3k = 60 \implies k = 20$

Nadia had $4k = 80$ dollars.

The parts total $2 + 7 + 11 = 20$, so $20k = 180$ and $k = 9$. The largest is the $11$ term.

$20k = 180 \implies k = 9, \quad 11k = 99$

Write $a = 4k$ and $b = 9k$. Then $3a = 12k$ and $2b = 18k$.

$3a : 2b = 12k : 18k = 2 : 3$

Silver is the $6$ term, so $6k = 18$ and $k = 3$. The total is all $5 + 6 + 4 = 15$ parts.

$6k = 18 \implies k = 3, \quad 15k = 45$

There are $45$ beads.

Apples minus plums is $8k - 2k = 6k$, so $6k = 36$ and $k = 6$. Pears is the $5$ term.

$6k = 36 \implies k = 6, \quad 5k = 30$

There are $30$ pears.

Write hardcovers as $2k$ and paperbacks as $9k$. The hardcovers are unchanged, so $2k : (9k - 200)$ equals $2 : 7$, and cross-multiplying gives the equation in $k$.

$7(2k) = 2(9k - 200) \implies 14k = 18k - 400 \implies k = 100$

Hardcovers is $2k = 200$.