*328Where the record indicates that both an absolute divorce and a divorce from bed and board were granted, plaintiff's argument that the trial court improperly granted a divorce from bed and board is moot. The trial court did not abuse its discretion in its award of child custody and child support where the trial court made findings of fact and conclusions of law in support of its decisions. Where plaintiff failed to rebut the presumption that a depreciation in certain property was not divisible, the trial court acted within its discretion to classify the depreciation of certain property as divisible.
Plaintiff Patrick B. Oltmanns ("plaintiff") and defendant Babette R. Oltmanns ("defendant") married on 21 December 2001. Two minor children were born of the marriage.
On 28 December 2010, plaintiff filed a complaint against defendant for equitable distribution, post-separation support, alimony, child custody, child support, and attorneys' fees. Defendant filed her answer, *329defenses, counterclaims, and motions requesting, inter alia, divorce from bed and board, equitable distribution, child support, child custody, post-separation support, alimony, attorneys' fees, Rule 11 sanctions against plaintiff, and a temporary restraining order and preliminary injunction against plaintiff.
Plaintiff failed to reply to defendant's counterclaims. However, plaintiff filed a motion for judgment on the pleadings concerning defendant's counterclaim for divorce from bed and board. On 20 February 2012, the trial court entered an order "grant[ing plaintiff's motion] in favor of Defendant." The trial court also noted in its findings and conclusions that, as plaintiff had not filed a reply to defendant's counterclaims, the allegations in those counterclaims would be deemed admitted. That same day, the trial court entered orders granting defendant's motion to compel discovery and awarding temporary child custody and attorney's fees to defendant.
On 7 November 2012, the trial court entered an order for permanent child custody, granting primary custody to defendant and secondary custody to plaintiff. On 31 July 2013, the trial court entered an order and judgment for permanent child support and equitable distribution, and reserved judgment as to each party's claim for attorney's fees concerning child support. Plaintiff appeals.
Plaintiff raises four issues on appeal as to whether the trial court (I) erred in granting judgment on the pleadings as to defendant's counterclaim for divorce from bed and board; (II) erred in its child custody ruling; (III) erred in its award of child support; and (IV) erred in classifying the post-separation depreciation of two houses as marital property.
Plaintiff argues that the trial court erred in granting judgment on the pleadings as to defendant's counterclaim for divorce from bed and board. We disagree.
Plaintiff contends the trial court erred in granting judgment in favor of defendant on plaintiff's motion for judgment on the pleadings as to defendant's counterclaim for divorce from bed and board. Plaintiff's argument first claims error based upon the trial court's deeming the allegations of defendant's counterclaim as to divorce from bed and board to be true based upon plaintiff's motion for judgment on the pleadings, since N.C. Gen.Stat. § 50-10(a) provides that the "material facts" of the *330counterclaim are "deemed to be denied ... whether the same shall be actually denied by pleading or not [.]" N.C.G.S. § 50-10(a) (2014). Plaintiff further argues that defendant's counterclaim for divorce from bed and board became moot after rendition and that the trial court should not have entered judgment upon the claim for divorce from bed and board because, in the time between rendition and entry, the parties had been granted an absolute divorce. We address plaintiff's second argument first as we find the issue of mootness to be dispositive, although not exactly in the manner as claimed by plaintiff.
The trial court heard the motion for judgment on the pleadings on 3 January 2012, *351and rendered the ruling in open court that same day. Judgment for an absolute divorce was entered on 16 February 2012. The written order granting judgment on the pleadings for divorce from bed and board was entered on 20 February 2012.
Plaintiff's argument that the 20 February 2012 order is moot highlights the difference between "rendition" of judgment and "entry" of judgment.
Under Rule 58 of the North Carolina Rules of Civil Procedure, a judgment is entered when it is reduced to writing, signed by the judge, and filed with the clerk of court. Announcement of judgment in open court merely constitutes 'rendering' of judgment, not entry of judgment. Entry of judgment by the trial court is the event which vests jurisdiction in this Court.
Although the written entry of judgment is the controlling event for purposes of appellate review, rendition is not irrelevant. The determination of mootness is made at the time of rendition, not entry of judgment. A trial court has an affirmative duty to enter a written order reflecting any judgment which has been orally rendered; failure to enter a written order deprives the parties of the ability to have appellate review. See In re T.H.T., 362 N.C. 446, 456, 665 S.E.2d 54, 60 (2008) ("[A] failure to proceed to judgment within a reasonable time deprives the parties of an adequate remedy at law, including the right to appeal a judgment entered. This Court does not have the authority to tell the trial court what judgment it should enter. We do, however, have the authority and the obligation to require the trial court to proceed to judgment when judgment has not been entered within the statutory time lines."). If a trial court fails to *331enter a written order, a party may apply to this Court for a writ of mandamus to compel entry of an order. See N.C. R.App. P. 22 (2014); see also In re T.H.T, 362 N.C. at 454, 665 S.E.2d at 60 ("In Stevens v. Guzman, [this] Court ... concluded that a writ of mandamus is the proper remedy for a trial court's failure to enter a written order." (citation omitted)). When the trial court rendered judgment on the divorce from bed and board, no absolute divorce had occurred. When the order based upon this rendition was entered, the absolute divorce had been granted and the divorce from bed and board no longer had any practical use. But if we were to accept plaintiff's argument that the trial court should not have entered a written order based upon a rendition which occurred prior to another event which rendered the claim moot, this would essentially eliminate the possibility of appellate review of a trial court's decision which was rendered prior to an event which might moot the order. Thus, the trial court must still retain the authority to enter an order accurately reflecting the judgment previously rendered, if only to permit appellate review of the trial court's action. Perhaps this is the meaning of the statement in Roberts v. Madison Cnty. Realtors Ass'n, Inc., that "[a] case is 'moot' when a determination is sought on a matter which, when rendered, cannot have any practical effect on the existing controversy." 344 N.C. 394, 398-99, 474 S.E.2d 783, 787 (1996) (emphasis added) (citation omitted).
Plaintiff's argument also highlights the fact that the order granting divorce from bed and board is now moot for purposes of appellate review.
Whenever, during the course of litigation it develops that the relief sought has been granted or that the questions originally in controversy between the parties are no longer at issue, the case should be dismissed, for courts will not entertain or proceed with a cause merely to determine abstract propositions of law.
Unlike the question of jurisdiction, the issue of mootness is not determined solely by examining facts in existence at the commencement of the action. If the issues before a court or administrative body become moot at any time during the course of the proceedings, the usual response should be to dismiss the action.
In re Peoples, 296 N.C. 109, 147-48, 250 S.E.2d 890, 912 (1978) (citations omitted). In addition, as no motion to dismiss for mootness *352is required, this Court can dismiss this issue on appeal ex mero motu. See State ex rel. Rhodes v. Gaskill, 325 N.C. 424, 426, 383 S.E.2d 923, 925 (1989) ("As *332no motion to dismiss for mootness has been filed herein, as is usually the case, we dismiss the appeal ex mero motu. ")
In the instant case, there is no need for this Court to consider the interaction of N.C. Gen.Stat. § 50-10(a) and a motion for judgment on the pleadings under N.C. Gen.Stat. § 1A-1, Rule 12(c) because our ruling on this issue "cannot have any practical effect on the existing controversy" and would be a determination of an "abstract proposition[ ] of law." Roberts, 344 N.C. at 398-99, 474 S.E.2d at 787. Plaintiff has not made any argument as to how the trial court's determination as to divorce from bed and board has had any effect upon the remaining issues which are in controversy in this appeal, specifically, child custody, child support, and classification of post-separation depreciation of real property. None of the orders relevant to these issues which are under consideration on appeal rely upon or reference any findings or conclusions from the divorce from bed and board order. Now that the parties are absolutely divorced, the divorce from bed and board has no remaining legal relevance. "A divorce from bed and board is nothing more than a judicial separation; that is, an authorized separation of the husband and wife. Such divorce merely suspends the effect of the marriage as to cohabitation, but does not dissolve the marriage bond." Schlagel v. Schlagel, 253 N.C. 787, 790, 117 S.E.2d 790, 793 (1961) (citation omitted). Plaintiff has made no showing as to why it still matters, legally or practically, that the claim for divorce from bed and board was granted, nor has he put forth any arguments regarding collateral consequences that may arise from that order in the future. The parties are now divorced absolutely and there is no dispute regarding the divorce itself. Thus, as the issue regarding the order granting divorce from bed and board is now moot, we, therefore, dismiss plaintiff's appeal as to this issue.
Plaintiff argues that the trial court erred in its child custody ruling. We disagree.
When the trial court makes a determination as to child custody, it must "consider all relevant factors" and grant custody to the party who will "better promote the interest and welfare of the child[ren]." N.C. Gen.Stat. § 50-13.2(a) (2013). As such, a trial court's findings of fact are conclusive on appeal if supported by competent evidence, In re Peal, 305 N.C. 640, 645-46, 290 S.E.2d 664, 667-68 (1982), and the trial court's decision as to child custody "should not be upset on appeal absent a clear showing of abuse of discretion." Browning v. Helff, 136 N.C.App. 420, 423, 524 S.E.2d 95, 97 (2000) (citation omitted).
*333Plaintiff contends the trial court erred because it awarded primary legal custody to defendant and secondary legal custody to plaintiff, despite making findings that plaintiff maintains an active role in the lives of the minor children and should, therefore, have been granted a joint legal custody arrangement. In its order for permanent child custody, the trial court made findings that plaintiff has "continued to be involved in the children's education and medical treatment[,]" "has been largely responsible for arranging [the minor daughter's] carpool to and from school[,]" and has arranged "fun activities" for the minor children "[d]uring his time with [them.]" However, the trial court also made findings of fact that: the parties have disputed school choices for the children; plaintiff struck defendant in the face while defendant was pregnant with their son; plaintiff hid a recording device in defendant's bedroom without her knowledge or consent; plaintiff sued both of defendant's parents, who provide the minor children with childcare, and defendant's paramour, for alienation of affection; defendant "has handled the lion's share" of the minor children's educational and medical treatment issues; defendant "continues to encourage [plaintiff's] relationship with the children[;]" and that the parties can no longer communicate by telephone due to arguments concerning plaintiff's lawsuits against defendant's parents *353and paramour. Moreover, the trial court made the following findings of fact, based upon a co-parenting counseling session the parties underwent in June 2011:
27. "Given that [the] parents have some differing belief systems, values and priorities, there are numerous areas where they might disagree on what is best for the children. Ongoing tension between them over decisions about the children's upbringing would have a more damaging effect on the children than the unilateral decisions of either parent. If the children sense or perceive that their upbringing is a source of animosity between their parents, there is the danger that the children will internalize responsibility about the tension. They might also use the "power struggle" between the parent[s] as a way to bring their own leverage to the situation and manipulate one [or] both parents."
28. [The counselor's] comments to the parties regarding their struggle to make joint decisions summarizes the Court's concerns in that regard. Due to the lack of trust between the parents, the differing values and parenting styles between them, and the fact that both parents are *334extremely intelligent, the Court finds that the parties are unable to make decisions of significance for the children together and that the power struggle between them is more detrimental to the children than unilateral decision-making authority to one parent would be. As a result, joint decision-making authority is not in the children's best interests. [Defendant] has demonstrated her willingness to rise above animosity and foster the children's relationship with [Plaintiff] and to genuinely consider his point of view in making decisions for the children. Accordingly, it is in the best interests of the minor children that [Defendant] be awarded primary legal custody of the children and have the authority to make decisions of significance for the children in the event there is disagreement between [Defendant] and [Plaintiff] regarding [the] same.
The trial court mirrored these findings of fact in its conclusions of law, noting multiple times that "[Defendant] is a fit and proper person" to have and be awarded the "primary legal care, custody, and control of the minor children [,]" while "[Plaintiff] is a fit and proper person" to have the "secondary legal care, custody, and control of the minor children." As such, the trial court acted within its discretion in awarding primary legal custody to defendant, as supported by its findings of fact. Cf. Diehl v. Diehl, 177 N.C.App. 642, 646-48, 630 S.E.2d 25, 27-29 (2006) (remanding for further findings of fact by the trial court where the child custody order awarded primary legal custody to the defendant-mother despite also indicating that "[b]oth parties are fit and proper to have joint legal custody of the minor children[.]").
Plaintiff further argues the trial court erred in "improperly den[ying] each party's request for a parenting-time right of first refusal." However, plaintiff has provided this Court with no supporting guidance as to how or why the trial court was required to make such a finding and, as such, we decline to address this contention. See N.C. R.App. P. 28 (2014) (declining to address arguments on appeal which are not supported by case law). Moreover, as the trial court orally noted that it would not entertain a parenting-time right of first refusal as being in the best interests of the minor children, it was within the discretion of the trial court to not include such a provision, either to its granting or denial, in the child custody order.
In addition, plaintiff argues that the trial court erred in its child custody order by imposing "unsupported travel restrictions" on the minor children. Specifically, plaintiff contends the trial court's custody order *335"lacks factual findings supporting such restrictions," which requires plaintiff to maintain the children's German passports and defendant to maintain the children's American passports. Plaintiff's argument is baseless however, as a review of the record indicates that both parties requested this arrangement for the children's passports in their proposed child custody orders. As such, plaintiff complains on appeal of an action which he himself requested from the trial court. See Romulus v. Romulus, 215 N.C.App. 495, 528-29, 715 S.E.2d 308, 329 (2011) ("A party may not complain [on appeal] *354of [an] action which he induced." (citations omitted)). Plaintiff's argument is overruled.
Plaintiff contends the trial court erred in its award of child support.
In reviewing child support orders, our review is limited to a determination whether the trial court abused its discretion. Under this standard of review, the trial court's ruling will be overturned only upon a showing that it was so arbitrary that it could not have been the result of a reasoned decision. The trial court must, however, make sufficient findings of fact and conclusions of law to allow the reviewing court to determine whether a judgment, and the legal conclusions that underlie it, represent a correct application of the law.
Plaintiff argues that the trial court erred in its award of child support because the permanent support award "improperly determine[d] each party's gross monthly income." Specifically, plaintiff contends his total monthly gross income was over-assessed by $4.00. Plaintiff is correct in his assertion, as plaintiff's total gross monthly income should be $7,269.08, rather than $7,273.08, as calculated by the trial court. Although such a difference is trivial and does not change the trial court's determination of child support, we remand for correction of this $4.00 error. See Peters v. Pennington, 210 N.C.App. 1, 26, 707 S.E.2d 724, 742 (2011) (remanding for the trial court to correct a calculation error of $0.50).
Plaintiff also argues that the trial court erred in its calculation of defendant's gross monthly income. Plaintiff contends the trial court's consideration of "negative rental income" generated by a condominium held by defendant lacked sufficient evidentiary support. This contention is without merit. The evidence presented by defendant in her financial *336affidavits at trial indicated that the condominium's negative rental income fluctuated between 2006 and 2010, and that defendant listed this negative income as costing her $1,063.18 monthly in her most recent financial affidavit. As such, the trial court acted within its discretion in setting defendant's negative income level at $1,063.18 per month, as this figure was supported by the evidence.
Plaintiff further contends the permanent child support award "improperly determine[d] and apportion[ed] the children's reasonable needs and expenses." Plaintiff argues that the trial court's determination of child support, which orders plaintiff to pay 19% of the minor children's combined monthly needs, was erroneous because the expenses of plaintiff and defendant were not considered equally.
Pursuant to North Carolina General Statutes, section 50-13.4,
[p]ayments ordered for the support of a minor child shall be in such amount as to meet the reasonable needs of the child for health, education, and maintenance, having due regard to the estates, earnings, conditions, accustomed standard of living of the child and the parties, the child care and homemaker contributions of each party, and other facts of the particular case.
N.C. Gen.Stat. § 50-13.4(c) (2013).
In its calculation of child support, the trial court relied on the financial affidavits of both parties to determine each party's average monthly shared family and individual expenses. Although plaintiff contends the trial court erred in its calculation of defendant's expenses, defendant's evidence showed that defendant incurred significantly higher monthly expenses than plaintiff due to defendant having to pay the mortgages and maintenance expenses on the marital home and the vacation home. As N.C.G.S. § 50-13.4(c) requires the trial court to have "due regard to the estates ... of the child and the parties," it was appropriate for the trial court to consider defendant's increased expenses relating to the two homes in determining its award of child support. As such, plaintiff's contention that the trial court erred in not making an even 50/50 allocation as to child support is without merit. See Brind'Amour v. Brind'Amour, 196 N.C.App. 322, 329, 674 S.E.2d 448, 452 (2009) (where this Court's *355analysis determined that the trial court did not abuse its discretion in its award of child support where the trial court carefully reviewed all of the evidence before it regarding the children's standard of living and the reasonable needs and expenses of both parties, before affirming the *337trial court's decision which denied part of the defendant-mother's claim for expenses).
Additionally, plaintiff contends the trial court erred in its award of child support because the permanent award should have been made effective from January 2011, rather than from January 2013. However, it is well-established by this Court that a trial court has not abused its discretion where, based on the evidence before it, the trial court chose not to modify the effective date of a permanent award. See Zaliagiris v. Zaliagiris, 164 N.C.App. 602, 611, 596 S.E.2d 285, 291 (2004) ("[The] [d]efendant finally contends that the trial court abused its discretion in making his child support obligations retroactive only until 1 February 2002. Prior to the entry of the permanent child support order, [the] defendant had been ordered to pay temporary child support in a greater amount than finally ordered. [The] [d]efendant argues that the trial court erred by not using its discretion to set an even earlier retroactive date for his permanent child support obligation. We conclude that although the prior temporary child support order was subject to modification, the trial court did not abuse its discretion in failing to modify that ... order to set an earlier retroactive effective date for permanent child support. " (citation omitted) (emphasis added)). Plaintiff's argument is overruled.
Finally, plaintiff contends the trial court erred in classifying the post-separation depreciation of two houses as marital property. We disagree.
Plaintiff argues that the trial court erred because it failed to make any findings of fact as to why the depreciation of the two homes constituted divisible property. However, as plaintiff fails to cite any case law which supports his assertion, we decline to address this issue. See N.C. R.App. P. 28(a) (2014) (holding that arguments not supported by appropriate authority will be deemed abandoned).
Accordingly, we affirm in part and remand in part for correction of the $4.00 error in the calculation of plaintiff's gross monthly income.
AFFIRMED IN PART; REMANDED IN PART.
Judges STROUD and HUNTER, JR., concur.