CHAPTER XV.

MISCELLANEOUS PROBLEMS,

INVOLVING THE VARIOUS APPLICATIONS OF PERCENTAGE.

I. Sold a cow for $25, losing 163%; bought another and sold it at a gain of 16%; I neither gained nor lost on the two; what was the cost of each?

II.

1II.

3. 100%-16%%=83%=selling price.

4. $25 selling price..

A. 5... 83%=$25,

7. 100% 100 times $.30=$30=cost of first cow.
8. $30-$25-$5, loss on the first cow, and gain on
second cow.

[1. 100% cost of second cow.

B. 2. 16%

..

13.

gain. $5=gain.

5. 1% of $5-$.3125, and

=

[cow.

6. 100% 100 times $.3125-$31.25-cost of second

$30 cost of first cow, and

$31.25 cost of second cow.

Remark. Since I lost $5 on the first cow, and neither gained nor lost on the two, I must have gained $5 on the second cow. • 16%=$5.

I. There have been two equal annual payments on a 6% note of $175, given 2 years ago this day. The balance is $154.40; what was each payment?

100% a payment.

$175,

of $175 $1.75, and

6%-6 times $1.75-$10.50-interest for 1 year.
$175+$10.50 $185.50= amount before paying
the payment.

(4.)

(5.)

(6.)

II.

(7.)

[payment. $185.50-100% amount left after paying the 1. 100%

$185.50-100%,

2. 1% of ($185.50-100%)=$1.855-1%, and
3. 6% 6 times ($1.855-6%)=$11.13-6%-inter-
est for second year.

4. $185.50-100%+$11.13-6%=$196.63—106% ·
amount before paying the last payment.
5. $196.63 - 106% 100% $196.63206%
amount left after paying the last payment.

=

=

$154.40 amount after paying the last payment .. $154.40 $196.63-206%.

206%

$196.63-$154.40-$42.23,

1% of $42.23-$.205, and

100%

100 times $.205-$20.50—the payment.

III. . $20.50 the payment.

Remark. In this solution we are obliged to use the minus sign, -, which is no obstacle to the student of algebra, but to the student of arithmetic it may seem insurmountable. To avoid this sign, we give another solution.

100% the payment. Then

$154.40+100% amount of the debt at the end of
of the second year.

100% principal that produced this amount.
6%=interest.

106% amount.

(1.)
(2.)

-(3.)

(4.)

(5.)

(6.)

.. 106%

of ($154.40+100%)=$1.4566,3+58%, 100 times ($1.4566+58%) = $145.662 +94% amount at end of the first paying off the payment.

$145.66+94%+100%= $145.66

year after

+1948%

=

amount before paying off the payment: amount at end of first year.

1. 100% the principal that produced it.

5. 1% of ($145.66+19438%)=$1.37167+ 83.225%, and

2809

6. 100% 100 times ($1.371167+1.83953%) =

$1371163+183,
2809 329505%=the amount at first.

$175 the amount at first.

.. $1371167+183,53% $175.

183.953-0

2809

2809%=$371742
2809

1% $371743-183,953-$.205, and

(15.) 100% 100 times $.205-$20.50-the payment. III. ... $20.50 the payment. (R. H. A., p. 264, prob. 5.) Explanation.-$154.40-the amount after paying off the last payment. $154.40+100% amount before paying of the last payment, or it equals the debt at the end of the first year plus the interest on this debt for the second year. .. We let 100%= the debt at the end of the first year, 106% amount of 100% for 1 year. . 106% $154.40+100%. Then proceed as in the solution.

I. If a merchant sells of an article for what of it cost,

what is his gain %?

II.4. 291%

III.

cost of whole article.

of 100% cost of of the article. selling price of 4 of the article.

of 874%-selling price of of the article.

5. 116% 4 times 291% selling price of the whole

article.

6... 116%-100%=16%%=gain.

.. 16%=his gain.

I.

(Milne's Prac., p. 360, prob. 51.)

A merchant sold goods to a certain amount, on a commission of 4%, and having remitted the net proceeds to the owner, received 4% for. prompt payment, which amounted to $15.60. What was his commission?

(3.)

4% commission.

100%-4%-96%-net proceeds.

1. 1%

amount received for prompt payment.

2. $15.60-amount received for prompt payment. (4.) 3.4% $15.60.

II.

(5.) (6.) (7.)

III.

(8.) §

100% 100 times $65 $6500 cost of goods. 1. 100%

$6500.

(8.) 2. 1%=130 of $6500-$65, and

3. 4% 4 times $65-$260-his commission.

... His commission=$260.

(Greenleaf's N. A., p. 441, prob. 11.)

I. If I sell 30 yards of cloth for $132, and gain 10%, how ought I to sell it a yard to lose 25% ?

(1.) $132 selling price of 30 yards.

$4.40 $132-30-selling price of one yard.
100% cost of one yard.

10% gain.

100% +10%=110%-selling price per yard.
$4.40-selling price per yard.

.. 110% $4.40.

1% of $4.40-$.04,

100%100 times $.04-$4-cost per yard.

1. 100%

$4.

of $4-$.04,

3. 25% 25 times $.04-$1=loss.

(4... $4-$1-$3-selling price per yard to lose 25%.

III. ... I must sell it at $3 per yard to lose 25%.

(Stoddard's Complete, p. 206, prob. 9.)

I. A merchant receives on commission three kinds of flour ; from A he receives 20 barrels, from B 25 barrels, and from C 40 barrels. He finds that A's flour is 10% better than B's, and that B's is 20% better than C's. He sells the whole at $6 per barrel. What in justice should each man receive?

$6 selling price of 1 barrel.

$510 selling price of (20+25+40), or 85 barrels. 100% value of C's flour per barrel.

120%

value of B's flour per barrel.

1. 100% 120%.

120%+12%=132%=value of A's flour per barrel.
4000% 40 times 100% what C received.
3000% 25 times 120% what B received.
2640%=20 times 132%=what A received.
9640% 4000%+3000% +2640% what all rec'd.
$510 what all received.

(13.)

1% 640 of $510-$.52211, and

[received.

(14.)

(15.)

4000% 4000 times $.52311-$211149-what C 3000% 3000 times $.52311-$158172-what B

24

received.

[received.

(16.) 2640%=2640 times $.52211-$139141=what A

$13911-A's share,

241

$158141
=B's share, and
($21114 C's share.

(Greenleaf's National Arith. p. 442.)

I. of B's money equals A's money. What % is A's money less than B's, and what % is B's money more than A's?

3. 100%-75%-25% excess of B's money over A's.
4. 75% 100% of itself,

I. At what price must an article which cost 30 cents be marked, to allow a discount of 124% and yield a net profit of 163% ?

(1.)

(2.)

(3.)

(4.)

II.

III.

I.

100%-30%,

1% 1 of 30%=3, and`

16% 16 times-5-profit.

30%+5=35¢ the price at which it must sell to gain 16%.

1. 100% marked price.

2. 121% discount from marked price.
3. 100%-12% 87% selling price.
4. 35-selling price.

(5.) 5. ... 87+%=35¢.

7. 100% 100 times .40=40=marked price.

.. 40¢-marked price.

(Seymour's Prac., p. 203, prob. 4.)

Had an article cost 10% less, the number of % gain would have been 15% more; what was the gain?

II.

conditional gain %.

8. 100%-90%=(100—90) times 14%=1×100%-100%

[difference. 100%)X 100% =

[the actual cost.

12. 100% 9 times 15%=135%-selling price in terms of 13... 135%-100%=35%=gain.

... 35%=gain.

A literal solution.

Let S selling price and C-the cost. Then S-C-gain and

(R. H. A., p. 406, prob. 87.)

=rate of gain. S-C-conditional gain and

conditional rate of gain. .. SC, whence S-27 C-1.35 C. .. 1.35 C-C.35 C=gain. .. Rate of gain=.35 C÷C-.35=35%.

I. In the erection of my house I paid three times as much for material as for labor. Had I paid 6% more for labor, and 10% more for material, my house would have cost $3052. What did it cost me?