Ratio Problems: Practice

The shared term $B$ is $4$ in the first ratio and $6$ in the second, so scale each to the least common multiple $12$: $3 : 4 = 9 : 12$ and $6 : 5 = 12 : 10$.

$A : B : C = 9 : 12 : 10$

Write the numbers as $7k$ and $5k$. After subtracting $4$ from each, $(7k - 4) : (5k - 4)$ equals $3 : 2$, so cross-multiply.

$2(7k - 4) = 3(5k - 4) \implies 14k - 8 = 15k - 12 \implies k = 4$

The larger original number is $7k = 28$.

Green minus red is $5k - 2k = 3k$, so $3k = 12$ and $k = 4$. The total is all $2 + 3 + 5 = 10$ parts.

$3k = 12 \implies k = 4, \quad 10k = 40$

There are $40$ marbles.

Write $x = 7k$ and $y = 3k$. Then $x + y = 10k$ and $x - y = 4k$.

$(x + y) : (x - y) = 10k : 4k = 5 : 2$

Scale $B$ to the least common multiple of $3$ and $4$, which is $12$: $2 : 3 = 8 : 12$ and $4 : 5 = 12 : 15$, so $A : B : C = 8 : 12 : 15$. Then $A = 8k = 16$, so $k = 2$.

$8k = 16 \implies k = 2, \quad 15k = 30$

So $C = 30$.

Scale $Y$ to the least common multiple of $3$ and $6$, which is $6$: $2 : 3 = 4 : 6$, so $X : Y : Z = 4 : 6 : 7$. The parts total $17$, so $17k = 3400$ and $k = 200$.

$17k = 3400 \implies k = 200, \quad 7k = 1400$

So $Z$ receives $1400$ dollars.

Write water as $9k$ and concentrate as $2k$. After adding $25$ liters of water, $(9k + 25) : 2k$ equals $7 : 1$, so water is $7$ times the concentrate.

$9k + 25 = 7(2k) \implies 5k = 25 \implies k = 5$

Concentrate is $2k = 10$ liters.

Largest minus smallest is $8k - 3k = 5k$, so $5k = 30$ and $k = 6$. The middle number is the $4$ term.

$5k = 30 \implies k = 6, \quad 4k = 24$

Scale $B$ to the least common multiple of $5$ and $10$, which is $10$: $4 : 5 = 8 : 10$, so $A : B : C = 8 : 10 : 9$.

$A : C = 8 : 9$

The shared term $b$ is $3$ in both ratios, and $c : b = 4 : 3$ makes $c = 4$ when $b = 3$, so $a : b : c = 2 : 3 : 4$.

$a : c = 2 : 4 = 1 : 2$

Longest minus shortest is $8k - 3k = 5k$, so $5k = 40$ and $k = 8$. The whole wire is all $3 + 5 + 8 = 16$ parts.

$5k = 40 \implies k = 8, \quad 16k = 128$

The wire is $128$ cm long.

The shared term $B$ is $5$ in the first ratio and $2$ in the second, so scale each to the least common multiple $10$: $3 : 5 = 6 : 10$ and $2 : 3 = 10 : 15$.

$A : B : C = 6 : 10 : 15$